[Animal]

A burning smell indicates burning of some kind, so you might see some scorching although I doubt it, but it's worth a shot. If it's 300W then you should be okay. I seriously doubt that it's your hard drive, as it would still show some activity, ie: you see your BIOS POST at the very least.
Don't worry too much about poking around in the case, just don't have any kind of food or liquid around while your doing and make sure you aren't standing on carpet and touch a metal part of the case BEFORE touching anything else to relieve any static electricty you may have built up.
I'm assuming you have a running PC. You could always try swapping the power supplies to see if that is in fact the problem, but by the sounds of it your PC knowledge is somewhat limited.
A lot of newer ATX motherboards have an led built right onto the board to indicate whether it's getting power or not. If it's lit you've got power, if not....
I had a similiar experience a while back, but with no burning smell. Playing some NHL 2002, and computer just shut down, no warning no nothing. Couldn't even get it it to POST after that. Swapped the power supply for a 450 and bingo, no probs.