It probably has to do with heat. Once a computer is under load, it tends to put off more heat than when it is idle. Try taking the side panel off the computer and placing a room fan so that it blows air into the computer. If you can do these heavy tasks using proper ventilation, it is probably heat causing your problem.
If that doesn't help, it could be bad memory or something like that, but let's try the heat solution first and see if it fixes things. If so, it's time to go out and get some new case cooling fans!
Also, bear in mind hard drives get very hot too - make sure they are ventilated during your test as well since you mentioned that these crashes occur during heavy disk access too.