10 years for a HDD is much time. Your HDD is likely dying. Note that every time you power on your Mac Mini lowers the chance to save data and raises the risk of complete data loss. Do you have a backup? How important is the data for you?
If the data is very important to you and you have no backups, don't try to recover the data by yourself and take the Mac Mini to some professional data rescue service.
If you have a backup and/or are just interested to learn data recovery, then you can try booting your Mac Mini into
Target Disk Mode by pressing T. Usually a dying HDD will mount, but suddenly unmount. Therefore you'll make a copy of the dying drive to a disk image on a working drive and never work on the original drive. For doing that, there is data recovery software around. One tool like this is the free
ddrescue. My hint is to install and learn how to use such a recovery tool in advance on your iMac. Only power on the Mac Mini, if you're really prepared for data rescuing.
Alternatively to Target Disk Mode, you can remove the failing drive and try a block level copy to a new drive with a clone station. One like this
icy box or similar.