Well, it took me long enough to get back to you folks. Sorry about that, since you'd taken the time to offer me some good advice. Attribute the delay to a combination of real life drama and the horror of having to poke around inside this infernal box ;)
Okay, here's the current report: As previously stated, I'd already swapped out the power supply with a nice new one, to no avail. GX1_Man suggested stripping pretty much everything out of the system and trying it then. I did so, leaving only the video card, CPU/heatsink/fan, and a stick of RAM. Powered it up, and the same thing happened - no POST, nuthin'. Only difference was that in this configuration, I got absolutely nothing, no matter how many times I powered off and back on again. Previously, if I powered off and on often enough, eventually the thing would start to POST. It would hang before booting the OS at times, but with enough tries, eventually I'd get into XP.
After a number of depressing tries, I decided to just start experimenting for the *censored* of it. First tried reconnecting the optical drives, thinking that if I could get it to POST, I would try booting Ubuntu. Nothing. Then tried reconnecting the hard drives. This time, it would sometimes POST (after several tries), but then complain that it couldn't find a boot device.
Finally, I rewired the whole thing, except for stuff I wasn't using at this time anyway - an old internal modem and a video capture card. Kept RAM, display card, dual optical drives, dual hard drives, and the floppy. Started powering up and down and up again, as usual. Eventually, the thing managed to complete its POST and booted Ubuntu from the DVD-ROM drive. Weirdly, although the POST reported finding all of the drives, Ubuntu couldn't "see" the internal hard drives, only a 1394 external drive I happened to have plugged in.
Got out of Ubuntu, rebooted the machine, and started the "power up, power down, power up" drama until once again it managed to complete POST, and it found XP again.
So, I'm basically in the same place I was when this started, except that it's taking more and more restarts to ever get anywhere. I hate to think what this might be doing to drives and the OS and such (maybe nothing - I'm not claiming any expertise here), but I'm starting to wonder if this machine is heading for the not-that-old computer graveyard, so I figured it couldn't do much real harm.
If y'all are up for it, I'd be grateful for any other thoughts/ideas/suggestions or interpretations. And maybe a little insight? Like, how come once XP is actually up and running, the machine works without a hitch, but once it's shut down, this mess starts all over again? If it's sick enough to refuse to POST and/or boot most of the time, shouldn't it stay sick and never do so? If it's the motherboard, is there any way I can replace it and the CPU (can't replace the items I have, since they're out of production - I've even scoured eBay for some) and still use my installation of XP, or am I doomed to a complete reinstallation?
Anway, there's a bunch of questions and a problem that remains to be understood. Thanks in advance for any further thoughts you all might have. It was nice to know that someone cared enough to respond in the first place.
I'll try to be more prompt in reporting back - assuming I'm able to get the computer to let me go online.