This could be several different things. First things first, turn it off and remove the power cord. Open up the G4, and find a little button on the logic board that looks like this:
Hold that button down for 10 seconds. Plug the power cord back in. Then start the machine up while holding down the Command(Apple key) + Option + P + R keys. It's a bit of a pain to hold all 4 buttons down at once. Keep holding that key combination down until you hear the startup chime twice, then let go and see if it starts up.
If that doesn't fix it, it's likely a power supply or logic board issue. If it's just a power supply, I've got some extras laying around that I'd be happy to donate to you. If it's the logic board, you're better off getting a new machine.
Let me know if you have any luck......