Mike,
Take the side off and first of all, check all the connections are solid. If its still not working, start unplugging components systematically and turning on again each time. Yes i've had a failed PSU before, but that came with a distinct smell, and no brief whirring sound.
The last time i had the issue you describe, it turned out to be a failed hard disk.
So, you can remove the following restarting every time:
CD/DVD-ROM
Floppy
Hard Disk
Individual Memory modules
Graphics Card
Even without these items, your computer should still POST - which is the bootup procedure. If you remove one and you get further, then thats your culprit.
Lastly, flip out your bios battery and leave for 30 seconds. It could even be something as nasty as a virus affecting your bios.