Strunk and White baldly assert that this is an error. They simply say don’t use they with syntactically singular antecedents like somebody. They don’t give a reason; and it is pretty clear they didn’t know anything much about the literary evidence that they has been grammatical and normal with singular antecedents for six or seven centuries. Strunk and White are just wrong about Standard English syntax, here as nearly everywhere else where they deal with grammar in their book The Elements of Style.
Obama is a fluent and excellent speaker of Standard English, and his grammatical ear (if not his political ear!) was spot-on perfect on this occasion. Singular they was the right pronoun to use in the context. If you talk about arresting a man when he’s in his own home, you’re talking about arrests of males; if you talk about arresting a woman when she’s in her own home, you’re talking about arrests of females; if you talk about arresting a man or woman when he or she is in his or her own home, you’re talking like a badly written statute or contract. Obama intuitively understood how to avoid all three of those undesired outcomes.
[Language Log reader Michael Straight tells me he once heard a clip on NPR of George W. Bush talking about the need for a father to take care of "his or her children." That would be how someone might put it if they felt anxious about committing a sexist blunder but didn't have the good sense to use singular they. —GKP]
This article does my heart good. I can’t tell you how much needless consternation over my commitment to “they” which I have felt over the years. It is the only sensible, efficient, non-sexist way in our language to refer in pronoun to an unspecified person and so has always seemed to me the obvious choice. But it’s such a landmine when regardless of its defensibility you know it will be perceived as poor form nonetheless.
Just a few months ago Pullum’s systematic take down of Strunk and White in an anti-celebration of their tyrannous treatise’s 50th Anniversary freed my mind from numerous other needless anxieties brought about by their shoddy and capricious thinking about grammar and style. If you haven’t read it yet, I highly recommend it.