My iMac G5 2GHz has been suddenly shutting off recently.
It also did the sudden fan spinning up thing last week.

From what I've read, it may be the battery, the power supply, or even the mother board.

I've zapped the PRAM and reset the SMU to no avail.

I wouldn't mind buying a battery, but I don't want to keep buying parts for a machine that isn't worth it, y'know?

Not sure what to do next...

Locate your Apple Hardware Test disc and let it run for a while in loop (control-L) mode.

I found that our work g5 would suddenly start the fan up full speed then die, had this for about a month then I moved it to allow more air round the case [it was jammed up against a cup'd]and cleaned the vent and the fault has not returned.

my 20" eyesight imac g5 periodically shuts down and i am almost certain that it is because of the heat and circulation. i took it into apple and the answer would have cost more than the machine. i noticed a large clump of dust (dirt) in the vent next to the power plug. took it out and the machine worked. problem occurred a couple of months later. same cure.

would love to clean the whole insides out but the back seems really hard to remove. it only has two (2) screws holding the ram access door not three (3) screws.

It sure sounds like an over-heating issue to me.