Ryan McAvoy said:

Before ditching the hardware I'd reccomend a clean sweep reinstall of windows and all your drivers and programs. Usually sorts out most catastrophic problems.

That's the first thing I'd try in this situation just to rule out hardware issues in case its a software issue.

Warbler, are you experiencing any blue screens of death (BSOD) before the restart? That usually points towards a hardware problem, which could mean a driver issue (and the newest drivers aren't always the best to have!), or it could be a physical problem. An old power supply can cause these kind of issues due to capacitor ageing, but I've seen similar symptoms from faulty hard drives, although usually you get a BSOD.

Installing Windows 8.1 doesn't seem wise on a 10-year old system. There's no guarantee it would even work because I believe it was designed to work with certified hardware, which a 10-year old motherboard probably isn't. Illegal operating systems should not be trusted, because you cannot guarantee security, or that it might sometime trigger the Microsoft Product Activation DRM and lock you out until you buy a legitimate product licence.

Do a clean install if you can, and put some decent free antivirus/anti-spyware software on. If you system starts behaving normally, then I'd put a bit of money up for a quality antivirus product, as the best free antivirus products cannot match the best commercial ones, IMO.

Edit: Bingowings suggestion is a good one. Check your power settings in control panel.