Power problems

You have a couple of options, replace the iMac. It is past due for a replacement and is considered obsolete by Apple, cannot run current versions of Mac OS, parts are not being made for it (except RAM and HD's) and even Apple Stores will not work on it because the parts aren't available.

Or option 2, is take it to your local Apple Authorized Service Provider to be diagnosed (likely a fee) and then fixed if parts are available.

It's really that simple. However, always use If your Mac won't turn on first to see if any of these items work.

Unplug the power cord from the back for 10 seconds and then plug back in. Try the power button then. If that works and is the only way you can boot up after shutting down you have a failed power supply.

Can we get a bit more information, please? Are there any signs of life? Fans spinning? Chimes? Flashing of the screen?

Holding the button may not be helping. My old 2010 iMac needed only a short press of the button to start---right tup to the day in late 2018 when it proclaimed its death by not doing anything.

If there are no signs of life, try a different power cable. Any standard computer power cable works. Do other devices plugged into the same socket as the computer work normally?

If you hear a chime, immediately press the SHIFT key to see if it will finish booting in Safe Mode. It is not a fix because Safe Mode is more of a "limp-home" expediency that makes some file checks and may help get it running in a very basic state. A successful Safe Mode boot when a regular boot fails commonly indicates that some of the system software is corrupted. If you get a successful Safe boot, then try a normal startup to see if Safe mode fixed something.

NOTE: Safe Mode boot takes a while. You need to continue holding the SHIFT key down until you see an on-screen message that yoou are in safe mode, or have a working desktop.

If it shuts down on it's own in the future first try the power button and if it doesn't do the job try the unplugging trick again. If that works then you have a power supply problem. Went thru the same on a 2014 27" iMac. But it had failed to the point where it would crash after only a few minutes running.