I have an older CompUSA desktop PC with Windows XP that has most of my pictures and files on it. The problem I'm having is that it shuts down completely random (acts like power is cut, just shuts off)

It won't reboot unless it sits "off" for about a week, and it's only good for one start when it does.

When it quits, and I try to start it up, the lights come on and it sounds like it's starting up, then the light on the Floppy drive comes on and it makes a buzz like a wrong answer on a game-show. :o ...let it sit for a week, and it starts up just fine, and works for a while.

Hopefully next week, it runs long enough to get all my pics onto my flash-drive to back-up on the laptop. :clonk: Last week I got all the wife & kid's pictures backed up, but didn't get mine done before it quit. :clonk:

Any idea what's causing it, and if there's any easy/cheap fix for it?

Sounds like a hard drive or the power supply about to quit.
You might try reseating all the connectors to the drives (including the power ones)and reseat the CPU and memory.
Leave the cover off and clean out any dust/debris ,then start it without the cover.
I'd start there.

Make sure your cpu fan is running, and its heatseak is clean. Although, it shouldn't take a week to recover from overheating.

Try putting the HD in the freezer for a while to extend the amount of time you get to recover it.

Keep in mind that putting the drive in the freezer can lead to condensation and finish it off.