Mac min 2014 won't start up

Try booting into Safe Mode in case some third party software is causing a problem.

Try running the Apple Diagnostics to see if any hardware issues are detected. If your Mini is using a hard drive, then there is a good chance the drive is failing. Unfortunately the diagnostics don't detect most drive failures. Let me know if you would like instructions for creating & using a bootable Linux USB stick to check the health of the drive if none of the other suggestions help.

Did you try to use Recovery Mode to do that, as explained in this article?

About macOS Recovery on Intel-based Mac computers - Apple Support