This could be anything from failing RAM (memory sticks) to a power supply unit which isn't putting out enough electricity. It could even be a dying motherboard. Open up the case and press the start button. Carefully put a finger on the hard disk to feel for any signs of life - a slight vibration. Check that the fans are running.
Power down and remove the RAM stick and if you have more than one, try each alone to see if you can isolate one which isn't working. Also try swapping slots - sometimes the slot is at fault. Examine the motherboard for blown capacitors - the little black cylindrical towers. If any are swollen and distorted from a perfect cylinder, they've gone west (south in the US ) and the motherboard is dead.