Oh dear
Some early iMac G5's suffer with one of three issues - swollen/leaking capacitors on the MLB, a defective power supply OR both issues
The power supply is cheap and easy to replace but the MLB - Well forget it, just not worth the hassle and expense of replacement or repair.
Apple did have a repair extension program in place for these issues but that was for units that were within 3 years of the purchase date and certainly less than the 5 years that this iMac is.
The REP for iMac G5 is no longer on the Apple website so that is not an option any longer.
That model iMac G5 is really easy to get into - You need a large phillips screw driver to undo the crosshead screws on the underside of the enclosure (perforated). Lay the unit display side down and loosen the screws which do not come out completely. Lift the back off starting from the bottom and swivel the rear cover up and away.
On the MLB, there are clusters of capacitors near the bottom and next to the CPU ducting. If any of these capacitors are 'domed' or have the appearance of a pyramid at the top, then the MLB is shot. Symptoms will be random hard shutdowns, failure to power on right away, wake from sleep, issues reading the HDD, garbled video (in extreme cases) OR screwed SMU. These problems may take a while to appear and then will often be reproducible as soon as normal operating temperature is reached.
If the capacitors on the MLB are in good shape, then the power supply is either clogged with dust (a flaw of this design) or on its way out. It may benefit from a good dusting though (no need to remove). A replacement PSU is surprisingly cheap. It is easy to remove although a tight fit.
Replacement of the rear cover is a little fiddly and you need to ensure it fits flush all the way around before tightening the screws...