[HoudiniMan]

If your computer had no hard drive at all it would still turn on fine. It wouldn't be able to load windows, but it would turn on and stay on until you turned it off. Hard drives can't cause the computer to shutdown at random, unless there is a short on the power connector (which a) I've never heard of happening and b) still wouldn't really be the hard drive).

Your problem sounds heat related.

The first thing you want to check is go into "Setup" aka the BIOS. Look and see if there is any kind of hardware monitoring or temperature readings for your processor. If you have a windows-based program from your motherboard manufacturer, try that - the others like SpeedFan are okay, but frequently report incorrect readings. I've had SpeedFan tell me my CPU was running -125 C.

Common motherboard safety limits for processor temperature are between 65-90C. Discarding the weirdo chips, you usually will see safety shutdowns at 70-80C. Your cpu temperature shouldn't be higher than 60C - if it is you need to improve your cooling.

Don't keep your computer in a cabinet or desk with no vents.

Worst case scenario, touch your heatsink with your finger. It should be warm because it's dissipating a good ammount of heat, but it shouldn't be super hot though. Check your fans, I had my CPU fan go out literally today, which caused a safety shutdown.

It is also possibly your power supply, but those are generally a-okay or completely dead - in fact (while not technically impossible) I've never seen one that worked "for a while". There are two ways to test a power supply unit (PSU): 1) Buy a PSU tester, it has lights for the proper voltages and you plug it into your PSU. 2) Replace it and see if it solved the problem. PSUs tend to run ~60 bucks and most stores have a decent return policy if you determine it's not the problem.

Lastly, I'm assuming it turns off - dead, no lights - and that it isn't crashing - not giving some kind of error or bluescreen. Correct me if I'm wrong.

FYI - I know what I've talking about. I'm Microsoft certified, worked repairing computers for the general public in retail for 2 years and saw a LOT of really out-there problems. Now I'm making my living as a career IT guy.