I took the machine apart this morning, got rid of some dust, jiggled a few wires--and now it starts up.
First thing I'm going to do is get an external hard drive for backups. For a couple years I was pretty diligent about backing important files up to dvd-rw's, but it's a pain and I got lazy. Connecting a USB hard drive isn't such a pain.
I've been considering getting a new computer anyway; that bullet I just avoided might be what gets me over the hump.
But I'm still getting an external hard drive.