*WARNING* Mac Errors
I don't know if Mac Minis have a recovery mode like the rMBPs do. Did yours come with an OS disc? If yes, you can boot off of the dvd and try and see if you can fix anything from there. If not (or if it doesn't have an optical drive) then chances are there's a built in recovery partition which you can boot off of. IIRC you hold down ALT while booting up and you can select which device to load up.
To restore your personal files, if you know someone else with a mac get them to come over (or bring yours to their place) and connect the two via Firewire or Thunderbolt, have their computer powered on and logged in and after they're connected, hold down "T" (you need a keyboard plugged into your Mini and must hold down the key on that keyboard) as your mini boots up. It should show up as an external hard drive on their end, you can back up your files that way.
That same person's mac could be used to create a bootable installation of OS Sierra too, if I'm not mistaken. You would need to format your drive though, hence why you'd wanna do the target mode backup thing.

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