Faulty Motherboard or Power Supply?

[Layback Bear]

I would recommend not assuming their is only one problem with a system that old.
Ten years plus is a long time for hardware to live.

 

[Mellon Head]

If the processor was toast, wouldn't the PC shut down after a few minutes?

No. The CPU controls all of the shutdown timing and signalling, so the system would just sit there happily doing nothing with the fans spinning, much like it is now.

But as Layback Bear has suggested, there is probably more than one thing wrong with your system. Given that you have no beeps coming from it without a GPU or RAM installed (unless you don't have an internal speaker - do you have one?), something is seriously wrong hardware-wise. Without a GPU or RAM, it should beep like crazy on bootup.