Exact model of the PSU please. If you don't know it, there should be a label with it printed on the PSU somewhere.
Can't say much right now other than 320W should theoretically be enough for that system.
Also check CPU temperatures. They should stay below 80°C, or at the very least below 90°C all the time. But it sounds more like the PSU than anything else.
A PSU may also just reach the end of it's lifespan after several years, so it's not necessarily because it's to weak to begin with, but because it's getting too weak. Possible PSU output degrades over time.
Max load is supposed to be 135w
Psu is rated for 320
Well that's not much power but might be enough for a system with a low power CPU and no GPU.
A good way to judge if it's the power supply: Did you get a blue screen or an error/shutdown message?
Often a power supply issue will cause a sudden power loss with no shutdown/blue screen or other error message.