iMac 27 5 K won't start up, screen went black and won't restart

Try using the Reset SMC (power management) procedure. For a recent iMac, it's basically unplugging the iMac from power (and plugging it back) in a specific way, described in this document

Resetting the System Management Controller (SMC) on your Mac - Apple Support

A 27-inch 5K iMac probably uses significant power. Make sure you are not connecting its power cord to an over-crowded power strip (surge suppressor). You can connect it to a power strip (that's not the problem), but keep other devices connected to the same wall power outlet (like printer, additional display, external drives, phone charger, desk lamp, etc.) to a reasonable number.