I noticed when I added a new album to my queue, that all of a sudden I had over 1600 *weeks* rather than a few hours. Scrolling down, I see one track which Amarok reports as 271806:50:10. This track isn't shorter or longer than most everything else on that album, as reported by Konqueror, which reports the file size in line with the others too.

I've read everything about this subject in the forum, including an early post from me! This is an ogg file, so it isn't related to missing mp3 file headers.

Can it possibly an error in the ripping program? I don't understand the voodoo between the file size, which is correct, and the track length, which is outrageous! Where in the process is the length tagged?

It could still be a header error - this problem shows up in some mp3s too. Yes, it's usually a ripping error, but could also result from track corruption as well.
Do other players report the odd track length? And, if you start playing the track in Amarok, what happens? There was a little script floating around called (I think) mp3fixer, might even have been distributed in Amarok 1.4.x tarballs, that attempts to repair dodgy headers. Probably worth trying if  you can't replace the track.
Did you rip it yourself? If so, maybe try ripping it again.

KMplayer and Kaffeine both will play the track correctly, as Amarok does. However, they report the correct length of 3:06.

So is this a header error? If so, why is only Amarok reporting the extremely long length?

Yes, I ripped the track, and will re-rip. But I would like to understand where the error is occurring, since I occasionally see other tracks with the wrong length, if not this extreme.