Should you be able to use one of the possible choices in macOS Recovery
to access either or both internal hard drives, you could use the Utilities from
Recovery startup to tell which hard drive has what content; and also try to
see if you could use Recovery to see if the failed hard drive could be repaired.
• About macOS Recovery - Apple Support:
https://support.apple.en-us com/HT201314
There are specific details in what key combinations to use for different results.
And additional links within above article offer more detailed choices...
Per instructions in above page: "...The re-install macOS utility installs different
versions of macOS depending on key combinations you used while starting up.
For details see How to reinstall macOS - Apple Support."
If you have a Time Machine backup, that may be useful in restoring the system
to either a new replacement drive, (outside the Mac, in an enclosure; or by wire
USB drive adapter without enclosure) such as this example from newertech:
NewerTech Universal Drive Adapter USB Bare Drive... at MacSales.com
And if that's all that failed, a restored system on replacement drive could then
be installed into the Mac; this can be tested prior to take-apart of the Mac. A
hard drive replacement for original in this manner should be able to run Mac
by use of the universal drive adapter USB wire kit (it has power supply, etc.)
There may be another issue behind the Mac failure; perhaps different hardware.
What symptoms does the Mac mini exhibit? ~ It may need to be tested to see if
any original components on the logic board or perhaps power supply(?) failed.
Perhaps an authorized Apple service provider may be able diagnose other issues.
Good luck in any event...! 🙂