The fan in the power supply runs of an internal 12v supply rail. The computer runs off many different voltages and are controlled by the motherboard. If the motherboard isn't getting all its correct voltages it won't be able to control the fans properly. Similarly, if the motherboard is failing it won't be able to control the fans properly.
The first step is to try a replacement power supply. If you have a second computer available with the same kind of power supply (pay attention to the numebr and size of motherboard plugs), or a friend with one, then try this as your first step.
If that doesn't make the computer work then chances are that there is something more fundamentally wrong with the motherboard.
One common problem with motherboards and power is capacitors breaking down. These are small cylindrical components usually no more than 1cm high usually with a flat top to them.
When they break down the top becomes slightly domed:
You could try scouring your motherboard with a good light source to see if any of the capacitors have gone.