The computer gods have frowned up on me. My hand-me-down desktop is on the fritz and the underlying problem(s) and solutions(s) are non-obvious.
Earlier in the week, the computer was booting and starting to boot Windows before suddenly restarting.
So I repaired the WindowsXP installation using the install disk. Afterwards, things started fine, but for some reason Windows updates couldn’t install.
Last night the computer either turned itself off at some point, or turned off because of a power outage. Either way, as of this morning the initial problem had returned in a worse form. About 10 seconds into booting the whole system restarts itself. It doesn’t even boot far enough to allow a boot-from-cd option.
I’m guessing bad RAM and/or motherboard. Your thoughts?

I’d definitely start with the RAM. If you have multiple sticks of memory, pull them out one at a time and see if you can get it to boot. Failing that, before you try the motherboard, consider whether it might be the power supply. I built a Linux box that I’d accidentally equipped with an underpowered power supply. It worked fine for a few weeks, and then the thing stopped booting just as you describe. If your power supply has been damaged in a way that limits its output, that may explain it. So, after some RAM debugging, consider trying a known-good power supply.