There is a possibility the clock battery on the logic board may have finally gone dead.
They require replacement once they do, and their failure can result in startup issues.
•Mac Mini - Model A1176 Repair Guide - iFixit:
https://www.ifixit.com/Device/Mac_Mini_Model_A1176
{If you have a startup disc, the DVD with original OS X for that computer, you could try
and start using it as a source, then see if Disk Utility can help diagnose, test, or fix it.
Sometimes a failed hard disk drive can also be part of a startup problem. Could be an
error has occurred in the system startup section of the hard drive; or the drive itself
may have a damaged sector. An old HDD may need to be replaced if the booted DVD
version of Disk Utility cannot repair it. The Snow Leopard dvd should be OK for this.}
To open and replace the battery requires knowledge and careful action so as to not
damage wires and delicate components in the way; skill, tools & attention to detail.
And have the correct battery on hand, from a reputable source, that is not too old.
https://www.ifixit.com/Guide/Mac+mini+Model+A1176+PRAM+Battery+Replacement/1114
The situation requires troubleshooting, most of the basics and lesser known causes.
You may end up visiting an Apple authorized independent service agent whose ability
to test & repair increasingly old vintage or obsolete computers and older OS, may be
helpful. There is a matter of age, a cut-off, when Apple Store genius staff cannot or
will not be able to help with a failed computer. Experience helps, in troubleshooting.
Good luck & happy computing! 🙂