If, however, it is an item in a list, we need to ask what the point of the list is; and we find it in Ephesians 4:17-24:
So I tell you this, and insist on it in the Lord, that you must no longer live as the Gentiles do, in the futility of their thinking. They are darkened in their understanding and separated from the life of God because of the ignorance that is in them due to the hardening of their hearts. Having lost all sensitivity, they have given themselves over to sensuality so as to indulge in every kind of impurity, with a continual lust for more. You, however, did not come to know Christ that way. Surely you heard of him and were taught in him in accordance with the truth that is in Jesus. You were taught, with regard to your former way of life, to put off your old self, which is being corrupted by its deceitful desires; to be made new in the attitude of your minds; and to put on the new self, created to be like God in true righteousness and holiness.
Immediately after this, Paul begins listing precepts (starting with "Put off falsehood and speak truthfully"), so the most reasonable interpretation of the passage is to see each of the precepts as intended to be a specific example of how the Christians in Ephesus should "put off their old selves" and "no longer live as the Gentiles do". If we keep this in mind by the time we get to 5:21, our interpretation has to be (in part) this: Paul is looking around at the pagan world, and what he sees is a world in which husbands mistreat their wives, parents provoke their children to wrath, and masters threaten their slaves with beatings. He then turns to the Ephesian Christians and says, "Don't do this; instead, be transformed in Christ. Husbands, do not mistreat your wives, but love them as Christ loves us all. Parents, do not provoke your children, but bring them up in the Lord. Masters, do not threaten your slaves, but serve them in the recognition that you and they are both equally slaves of Christ, who plays no favorites." Pauls' fundamental message here is (and has to be, given 4:17-24) transformative: he is not re-affirming pagan roles but telling Christians to transform them.
Now, I think you can put forward a reasonable argument that it is unfortunate that Paul here stops short at transformation of roles rather than going on to tell the Christians to break those roles entirely. It's not really an argument with which I agree as far as Paul goes, in part because I think the examples here are precisely intended merely to be examples, and that the general principle here - that Christians in whatever role the conventions of society may put them should radically transform those roles "out of reverence for Christ" - is more valuable than particular instructions to eliminate this or that conventional role. An instruction to eliminate a given role is limited to that role, but the general principle given here applies all the way across the board, even to things that Paul couldn't possibly have thought of in his first-century context. Even if Paul is not going far enough, what he is giving us here is something that makes progress possible in any context, whatever the roles society is attempting to thrust on us. And it becomes especially important for people in positions of power (Paul never, ever says that husbands have any authority to subject their wives in any way; and a look at the master-slave sections shows just how radical Paul's general intent here is: it is entailed by what Paul says that master-slave roles are merely conventional, and their real role, in Christ, makes them equal; the parallelism here suggests that Paul is telling us entirely to reconceptualize all social notions of authority and power in terms of service and love). And, while I don't expect non-Christians quite to grasp the point at first glance, to tell a Christian to love someone as Christ loves the whole Church is to set an extremely high standard; for the Son of Man came to serve and to set free, and went so far as to suffer torments and give his life for that end.
Nonetheless, I think an even more reasonable argument can be made that other statements on these subjects in the Pauline epistles are less radical. 1 Cor. 11:7-10 gave Augustine considerable trouble when talking about how human beings are made in the image of God; Augustine was certain that women and men equally fall under this description, for Scriptural, traditional, and philosophical reasons, but was puzzled as to how Paul could then argue in the way he does in this passage. His solution in De Trinitate was that the author is here saying nothing about men and women in themselves, in Christian marriage, or in Christ (as is proven by what he goes on to say in verses 11 and 12); rather, he is taking a conventional view of marriage and using it symbolically. (The whole discussion is even more complicated, because Augustine was also puzzled about how in the world the angels in verse 10 are supposed to be relevant to the subject, and spend some time trying to figure that out.) So it's fair to say, I think, that there are some very obscure and puzzling things said on the subject (1 Tim. 2 is a case in point; I'd have to look it up, but I think a standard interpretation of the childbearing remark has been to see it as an allusion to Mary's childbearing - but it's certainly rather cryptic). But it seems (to me, at least) that what people really have a problem with is simply Paul's view that Christians, as a rule, should not cause scandal; because it is this, I think, that is most plausibly at work [at least in the background] in these cases. I think it's a defensible view, but I can appreciate that others might disagree with it; and it has to be reconciled with the example of Christ, who certainly was not an avoider of scandal. (Again, I think it can be done, but I can understand others not agreeing.)
At one point in the discussion, Schwyzer was asked what would be required for him to conclude that the Bible is sexist; I can't speak for Schwyzer, but my response to such a question would be that this sort of question implicitly works on an assumption that there is a clear line that can be drawn, such that everything on one side of it is sexist and everything on the other side of it is not. But this is not, I think, what's really at issue with sexism; it's a pervasive result of our fallen natures. If our fight against sexism consists entirely of identifying things as sexist and eliminating them, we will never have done, because human ingenuity will always come up with yet more ways to be sexist (or racist, or anything similar). What we need instead is to begin transforming the very nature of the human mind. Critique is not enough; to rely on critique alone is to guarantee defeat, because it binds you to perpetual reaction. Rather, we need the human heart to be transformed - and, in particular (from a Christian perspective) - we need the human heart to be transformed in Christ; because divine love cannot be shackled. The hard work of transformation is the only effective anti-sexism; critique needs to subserve that.
By the way, since I occasionally complain about people calling themselves 'progressive' when they clearly are not (as can be seen from their words and actions), I should take the opportunity to note that Schwyzer's calling himself progressive is no lie; he really is. I don't always agree with him, but the point of progressivism is general progress in patience, compassion, and wisdom, starting with ourselves, not perfection or present possession of them. (I must confess: halfway through that sentence I began trying to make as many words as possible start with the letter 'p'.) And the way we progress in these things is (1) self-critique; (2) practice; (3) interaction with others. And Schwyzer is in general as exemplary in all of them as we humans can reasonably be expected to be; one of the reasons I've put him on my sidebar is that I think his is a weblog that will stimulate me in new directions of progress myself.