Q: Powering Down?

Try a Safe Boot, holding SHIFT key at bootup… does it do any better?

Assuming you have already done the basics like reseting the pram I would look at overheating as a possible culprit. G4's had a little heatsink fan that can go bad. Your thermal paste may have gotten too dry too keep down temps. What model of g4 is it. Sawtooth,Quicksilver,  MDD. etc?

Your machine will have a barcode on the back of it that will tell you everything about it. In the meantime, boot the computer with the door open. Verify that both the big heatsink fan(bottom of the case) is spinning up fast and that the little heat sink fan( front of the case is also spinning.