Gary can't turn on his computer unless he unplugs it, waits for five minutes, then plugs it back in. Leo says that if Gary is turning it off with the hardware button, the computer may be going into hibernation and there are known problems with a system in hibernation. He should disable that first.
Leo also advises shutting the PC down from the start menu, rather than turning it off with the power button. It also could be a faulty power supply. USB devices may be the issue, according to the chatroom. Try unplugging all USB devices and see if the issue continues. He has three external hard drives, and there may not be enough power for those. A bigger power supply should solve that.
Another issue may be that the PC is looking to bootup from the external drive. He should see what the boot order is in BIOS. While he's there, he should check the BIOS Power off status.