In this chapter, the LORD establishes a ritual test for a wife's adultery.
I will begin by quickly addressing the first notes on defilement and restitution of sin and then discuss the "adultery test" from verse 11 and onward.
First is a quick note that everyone who is ceremonially impure must stay outside of the camp until they have become clean. This is an extension of the Levitical regulations, because previously only people with a skin disease had to live outside the camp (cf. Lev 13:46), but people with other types of impurity were only required to avoid touching other people. Now they are also required to dwell outside the camp, which is generally consistent with the theme of separating impurity away from the rest of the people.
The second is a fairly vague regulation that anyone who "commits any of the sins of mankind" (that seems pretty broad...) must make restitution "in full... and add to it one-fifth". This seems to presume that there is some sort of monetary value or assessment of the "sin" committed, so this probably only refers to situations where there is an easily assessable damage. Otherwise it makes no sense to say that a fine (for instance) should be increased by 20%: you could just as well make the fine 20% larger in the first place and then this regulation is moot. So this probably only refers to situations where somebody causes a certain amount of damage to another person and the 20% extra is meant as a punitive award to the injured party.
As for the "ram of atonement", the closest reference to this is Lev 5:16, which is to atone whenever someone "sins unintentionally against the LORD's holy things". Interestingly, that verse also mentions a 20% surcharge in addition to full restitution and the ram offering. In that case, the "holy things" probably means either tithes or other expected offerings that the person was supposed to give the Levites or the priests, which is why restitution in that verse goes directly to the priest. In this case, the restitution goes to "him whom he has wronged", which means that it generally refers to the population at large. The ram of atonement is probably meant to be the same kind of ram offering as Lev 5:16 however, which is a guilt offering. Lev 5:16 is specifically referring to unintentional sin, while this passage seems to include both intentional and unintentional, though it must not be a severe enough sin to warrant the much harsher penalties for e.g. theft (namely, Ex 22:1, 22:7, 22:9).
With that out of the way, I will now discuss the much longer and more substantive "adultery test". The first few verses establish that this is the ritual to use if a man suspects his wife of adultery but there is no evidence that would support a criminal prosecution (witnesses, caught in the act). However, the man has a "spirit of jealousy" (Hebrew, "ruach qinah") so that he suspects his wife of committing adultery. In this case, there is a fairly elaborate ritual involving a small "grain offering of jealousy", which seems intended to recompense the priest more than anything else, and then there is a scroll with curses that is washed off into the "water of bitterness" and the woman must drink it. This is a peculiar ritual, but the intent is pretty clear: it is meant as a ritual curse to cause infertility if the woman is guilty and nothing otherwise.
I have already discussed the importance of bearing children at great length, so I won't discuss it more here (see my commentary on Gen 14 for more discussion), but suffice to say it is still just as important to the women of Moses's day as it was in the time of Abraham. This is strange because there are very few ritual curses in the law of Moses, other than the curses for disobedience to the LORD (in e.g. Ex 20:5, Lev 26). I suppose we could view this as being a curse for disobedience in its own right, because the cultural expectation is that women would be obedience to their husbands (as the husband/father is the head of the household). It's just strange that there would be a ritual to perform only if the husband felt jealous, rather than the LORD promising to invoke justice upon anyone who commits adultery (which we saw in e.g. Lev 20:20-21 for certain types of incest).
So the strange thing here in my view is not punishing adulterous women with infertility, but rather the fact that this only occurs if her husband becomes jealous and submits her to the priest. Furthermore, even though the woman is brought to stand "before the LORD", the curse itself seems to be only vaguely related to the LORD and is more of an imprecation from the priest himself. This "adultery test" seems unrelated to nearly everything else in the biblical regulations and almost seems inconsistent: so much of biblical law is based on either 1) human punishment based on witnessed crime or 2) divine punishment for another set of crimes. In this case, it's not really divine punishment because it is initiated by the woman's husband and the priest, but it doesn't fit normal pattern for human punishment because there is no evidence or witnesses.
It seems to me as if this is an attempt to regulate or prevent men from harming their wives when they suspect marital infidelity. Maybe the reason there is concrete action (a grain offering, the scroll, the bitter water) is to give the husband something practical that he can do if he becomes jealous, rather than e.g. killing his wife, which would be murder under the law and punishable by death, but who knows if that were normal at the time?
The truth is that I don't know, but I do not believe this ritual is ever mentioned again in the bible, so there is nothing that we can cross-reference against to learn more about it.
Now that I've discussed some of the details of the ritual, I would like to take a moment and talk about the bigger picture of what this says about male-female relations.
Obviously, what it says is that the husband has a great deal more power and control than the wife. This chapter implies that men can have multiple wives (as we saw with Abraham and Jacob) and that women distinctly cannot have multiple husbands. This is a ritual that can only be undertaken by the husband because by definition the husband is not guilty of infidelity if he sleeps with another woman, because he can always have more wives. Of course, if he commits adultery by having sex with a married woman, then he would be guilty and punished by death, if he were caught.
The other imbalance of this chapter is that the husband can subject his wife to this ritual simply if he is jealous. That seems to violate the woman's due process, since there is no evidence or anything. Of course, this ritual is not exactly a punishment, but I can imagine the public embarrassment from even being accused would be pretty great, even if one were not guilty of the crime. And in all of this, "the man will be free from guilt, but that woman shall bear her guilt" (v. 31).
So those are the major imbalances of this chapter, how this ritual stacks everything in favor of the husband and against the wife. There is a small price to pay by the husband (having to make the grain offering), but there is otherwise no cost or penalty for a false accusation.
So the bible is pretty unfair towards women, yes? Well, maybe it is, maybe it's not. As with many of the regulations of slavery and other marital laws, we have to ask if the bible is instituting these injustices or containing and regulating even greater injustices that were common beforehand. I think any student of ancient Mideast history would find that 1) slavery was widespread and universally accepted, 2) male dominance and maltreatment of women was widespread and universally accepted. Both of these points are true generally outside of the bible, so it should be nearly self-evident that the bible is not the source of the cultural traditions that created these kinds of injustice.
Then we have to ask ourselves a question. Is the bible encapsulating and authorizing these injustices or limiting and restraining them? I think in some respects the answer is both, but I believe that the intent was the latter. That is, at the time the Pentateuch was written, provisions governing the treatment of slaves (addressed earlier in Exodus) and wives (such as this chapter, but also in Exodus) were probably meant to limit abuse and give additional rights to slaves and women. For instance, if a master injures his slave, the slave is given freedom as recompense for the injury (Ex 21:26-27): I think it's unrealistic to presume that absent such regulation that masters would have been even more kind to their slaves.
However, given the progression of history towards more equality between genders and the abolition of slavery, these historical provisions seem harsh and regressive. Furthermore, it is fairly indisputable that many people have used passages such as these as proof that slavery is okay or that men should control their wives and have many rights over them, emboldening those who commit injustice. But given what I said above, I don't believe that was the intent of these passages or of the "adultery test" here. What could have been a containment of injustice in the past is an expansion of injustice in the present, simply because the bible, once written, cannot be revised.
That leads to another two, inter-related questions. If the bible contains these regulations of injustice, then 1) is this evidence that the bible is not divinely inspired and 2) is this evidence that the bible is anachronistic and not relevant to our time? Many people answer these questions, "yes", but I think a more nuanced approach is both justifiable and appropriate.
To address these two questions, the answer depends on a very specific assertion about God's intent when inspiring the bible. That is, one must presume that the LORD is intending to only write laws (no matter the context) that are universally applicable across all time and cultural boundaries. That assertion is simply untrue, for a variety of reasons that are too lengthy to enumerate here. At the simplest level, this goes back to what I said about progressive revelation: God could, and does, reveal laws that are only true for some time and in some places before a proceeding revelation "overwrites" or obviates the prior revelation. There are some laws which are permanent and some laws which are temporary, and we just have to find out which type this is. Since the adultery test is specifically associated with the priesthood and the tabernacle offering system, this test must be temporary and a component of the Mosaic code that was fulfilled in Christ.
Another important point is that the Pentateuch was written as the national charter and constitution for the new Hebrew nation. This means that virtually everything which was to be a national law had to be written down, especially if it differed from the cultural norms of the time. If there was even going to be a law about women's rights, it would have to be written down in "The Law", because there is no separate royal constitution or law: the LORD is the king of the Hebrew nation, and this is his royal law.
Is the Law written for the Israelites, then, or for us? This gets back to the question of intent: what was the intent of the LORD when he inspired the Mosaic Law? Some people erroneously (and usually unconsciously) assume that the Law was only written for us, but that's not true. It was written for both the present time and the Hebrews of the past. When we read the Pentateuch, we see the text in the form it was given to the ancient Hebrews, as a legal and religious treatise to define their new nation, giving them a tradition they could rally around to unify the disparate and frequently quarreling tribes. But in application to the modern times and modern culture, it must be viewed through the transformative power of the NT, because the Pentateuch is not the last word of the bible, it's actually the first word. But just like a word is changed by its context, so is the meaning of the Pentateuch changed as it is placed in the larger story of the bible as a whole.
To summarize, we should view the Pentateuch (and by extension, this chapter) for what it is, a legal structure given to a small collection of nomadic tribes sometime in the first millennium BCE. Its relevance to them is by giving them a faith and a law to guide their lives and interactions on a day-to-day basis and that is what defines its structure. But we must also remember to view it as written to us, the first part of the greater biblical story, and that is what must define its meaning.