I finally got around to opening up the machine today, and I found what the problem was:

It's quite clear that at least 3 of the capacitors blew out. I'm not clear how this happened, I know that excessive heat can do it but the fans on the system work just fine and there was no excess buildup of dust when I removed the back cover. And I could understand one capacitor failing, but 3 at once? That just doesn't make any sense to me.
Oh well, the machine was old, and served me well for the nearly 6 years that I had it. I'll just stick with my newer machines for now.