The problem could lie in any of the major components:
Motherboard
CPU
GPU
Power Supply
RAM
If your computer refuses to start up and starts up again after a few hours or days, I'm betting on the motherboard dying out.
Your system is quite old now, if you game on a regular basis maybe you want to think about investing in a newer system.
If you don't want to or can't upgrade you can always take it down to a store and ask them to take a look at it, although if the problem cannot be replicated then you have a major problem since if no one can find what's wrong no one can fix it.
Edit: You can always stress test individual components to see what breaks first if anything.
Run PRIME to check your CPU
MEMTEST86 or SP2004 to check your RAM
Run CPUID to check if your voltages are optimal, and also search on google for stress test programs and run them one at a time.
If something hangs up on a particular test atleast you'll know what's wrong.
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