2006 imac randomly shuts off

Does it shut down completely (power off), or does it go to sleep with a black screen?

Mine was going to a black screen, but would come back to life if I waited a moment and hit a key. It turned out to be a heat issue. I used smcFan Control to ramp up the fans and it never happened again over the next 6 months.

It's free, so you have nothing to lose by trying it.

I remember my iMac was going to sleep but you couldn't wake it up unless you pressed the power button and sometimes you had to restart it.

Try repairing disk permissions, resetting the PRAM and SMU. Doing a web search will tell you how to perform these easy procedures.

Come to think of it it actually turned out to be a power supply issue. I moved my Mac into another room by itself and the problem went away. I think it could have been my power strip(s) setup and other devices. I moved things around to different outlets and that took care of the problem.

My sister had this problem and solved it with an uninterruptible power supply. Dirty power can be unkind to desktops.