If the Power supplies +12v rating was sufficient, I could almost give the go ahead. But of course a PSU struggling to provide power to the system, for lengthy periods of time, can have it's toll. PSU failure is the most likely thing, I wouldn't imagine it'd catch fire, but all is possible. Singular components will likely go unharmed, unless of course the failing PSU sent spikes through a failing voltage regulator, could fry the video card, motherboard, ram, CPU, etc.
On the other hand, the system in mind may not tax the 300w to levels needed for such disastrous outcome. Recommended current and wattage requirements are not minimum requirement. My 3800 x2, 7600gt, 2 gigs ram worked wonderfully on a 200w, +12@10a unit for 2 weeks, waiting for a replacement. So as you see, there are arguments either way, safest route is buying a PSU that provided sufficient power. But all things considered, I wouldn't be surprised if it the current configuration worked like a charm, providing no trouble.
Btw, doom may not put the system in a situation where it's drawing at peak levels. Take this into account.