Background: I took my Dell desktop out of storage today after about 5 months. It is almost 9 years old, running windows xp with a intel pentium 4. I don't know most of the specs but I think it could be a power source issue.

The problem is that the computer starts up fine, will load windows and after about 1-2 minutes it just shuts off with no warning. It got pretty dusty in storage, so I opened it up and tried to blow out most of the dust, but it didn't make a difference. I don't think it's an overheating problem because it doesn't feel any warmer than usual and I can turn it right back on after it shuts off and it tends to even last a few seconds more.

Any ideas/suggestions would be greatly appreciated.

Just try booting you Pc, with just motherboard and video card connected, unplug the rest.
It will load with a boot failure showing on screen, leave it like this and if there's another shutdown, you can be pretty sure it's a power supply issue.
If there is none, just plug back each device one by one , do the same test . I would suggest to connect hard drives and DVD drive last, it could be a problem for those devices.

What model Dell?

Can it boot into safe mode? There is a virus that causes shutdowns, mytob maybe?

Sounds like an overheating issue, but I'm no expert. Also sounds like time to buy a new computer.

Don't want to jinx it, but thanks, I did this (after figuring out what was what in my computer, lol), and it kept working in the boot failure mode, so I started plugging things back in one by one, and it's working with the cd and dvd drives unplugged. I'll figure out which one it is later, I'm just happy it's working now. Thanks again!

Edit: Just curious, how would the drive shut off the computer? Is it like an electrical short or something?

Try to boot in safe mode with your hard disk to see the failure still exists, if it fails, you can be pretty sure your hard disk drive has a problem.
(You might still try to repair it using manufacturer tools, you can find them on this CD : http://www.ultimatebootcd.com/ . If you have not data on it, run a low level format to see if it fails, then the repair tools).