The easiest method is to purchase a smaller computer that fits inside. If you do a google search for "atom pc" you'll find a lot of options of reasonably useful machines that will readily fit into the G5 case. You won't be able to use the original slots for cards, but you should be able to extend SATA cables outside the atom pc to the drives, and attach the power button wire to the correct spot on the atom PC. You can also extend the USB ports to the outside of the G5 case.
It's not ideal, but for a quick hack that's easier than a full PC upgrade, it's cheaper and easier if you merely want the design aesthetic.