Now in this case I have a clone of the drive from 20 days ago to which I can restore, eventually. The non-booting drive can be read, as usual; it just won't boot. I can't run FIXBOOT on it (in case that somehow was the problem because I've just discovered, the hard way, that this computer (which I bought secondhand) has an Administrator password on Windows, and I don't know what it is. (It's likely whoever sold it didn't know either.) So I can't run the Repair console on a WinXP CD-ROM.
I find it hard to believe that this is viral in nature because it's happened so many times, but I'm running anti-virus scans on the non-booting drive now. The backup has already passed. Because the backup drive is 20 days old, as soon as I booted it (leaving it connected to the internet) it wanted to install lots of updates, so I need to wait for it to be done with that as well before I start moving the data from the non-booting drive to booting one. What I guess I need to do also is to go buy another 320 GB IDE drive so that I can have two clone backups at once, because right now I can't make another clone of the booting drive without overwriting the drive that was working just fine this morning.