My 2008 iMac will not turn on

1) Reset the SMC by unplugging the iMac, waiting 15 seconds, then plugging it back in.

2) Try a different outlet and/or surge protector.

3) Listen for a startup chime. If you hear a chime it might be a display issue. No chime is a logic board or power supply issue.

4) Hold a wake celebrating 12-13 years of life before sending it out for recycling. Then buy a replacement.

If resetting the SMC lets you boot up again but the iMac shuts down again by itself it's a sign of a failing power supply. It would cost more than your iMac is worth to replace it. At the age of your iMac it's like giving a heart transplant to a 98 year old patient with terminal cancer.