Try resetting the PMU chip.
With the user access cover off the bottom of the base, and the airport extreme card on the 'left-hand' side, and the user accessible RAM slot on the 'right-hand' side, you'll see some test points above the airport extreme card. The PMU button is just above the test points. See the picture below.
Disconnect the power cord. Remove the user access panel from the bottom of the computer. With the user access cover off the bottom of the base, and the airport extreme card on the 'left-hand' side, and the user accessible RAM slot on the 'right-hand' side, you'll see some test points above the airport extreme card. The PMU reset switch is (hidden under the protective cover) is just above the test points.
Press the PMU reset switch (located under the protective cover, next to the test points) once on the bottom side of the logic board. Do NOT press the PMU reset switch a second time because it could crash the PMU chip.
WAIT ten seconds before connecting the power cord and powering the computer on. If the computer powers on, replace the user accessible panel. If the computer does not power on, there is something else wrong with the computer. Try these things.
If there is no power, no fan, no hard drive noise, and the screen is black, then
verify the power outlet is good. Plug a different device into the socket to ensure there is power, or plug computer into another outlet.
Check the power cord. Use a known good power cord. Check connection of the power cord on both ends. Verify that the plug is securely plugged into both the A/C outlet and back of the computer.
Remove keyboard, mouse, and other peripherals such as speakers. Disconnect the power cord, place the computer in the service stand, and remove the user access plate.
Using a voltmeter, check the voltage on the battery test point. Place the positive probe of your volt meter on the battery point, and the negative probe on the growing point. If the reading is over 3.5 volts, the PRAM battery is okay. If the reading is under 3.5 volts, replace the PRAM battery and test again.
Warning: Whenever the bottom housing is opened for service, you must do two things:
1) You must clean the original thermal film from the surfaces joining the thermal interface layer and reapply thermal paste to the thermal pipe.
2) The bottom housing has four torx screws that must be tightened to at least 17 in.-lbs. If you do not have a torque driver, you will have to make sure these screws are tightened by hand FIRMLY, BUT NOT FORCIBLY. If the bottom housing is not securely attached to the base in this fashion, the CPU may overheat and become damaged.
To learn how to replace the PRAM battery and do other testing, go to scribd.com, and search for the iMac G4 Repair Guide.
I took the computer to the apple store. They did some basic diagnostics at the genius bar, then told me that it was no longer repairable because it was too old. Although they did say that it was probably a power supply. I have been reading on line that when the PRAM battery is below a certain voltage the computer will not start. Do you know if this is true? It seems a little crazy that the computer would not start because of the PRAM battery. But it would make sense in this case besause the computer always had the wrong date and time. I just never thought that it would not start because of it.
What are your thaoughts.
I would recommend testing your power-cord with an ohm meter to make sure the power cord is okay. Thee was a thread on here a mont ago or so from someone looking for a power cord for the G4 (don't remember is their cord was bad or missing). The point is, they can go bad. It's free to do the test.
Everything depends on how much you're willing to spend. Replacing the PRAM battery is inexpensive. And that could actually be the cause of your troubles. If you decide to replace the PRAM battery, while you have the computer open, test the power supply as shown in the iMac G4 Repair Guide.
I would look into the cost of a new power supply, or a new inverter board, or a couple of other things, and then decide how much you're willing to spend before you start replacing items to fix this computer.