Almost every time, you cannot run an older Mac OS on a newer machine. There are *EXTREMELY* few cases where it is possible.
That said, when you say "2014 Mac Mini," do you mean "one purchased in 2014 that you are going to buy used" or do you mean "the currently available Mac Mini"? I assume you mean the currently available model, also called "Late 2014".
If that is the case, then you're out of luck - I know of no way to get Mavericks running on it. If you're okay with running Yosemite, and are going to buy it anyway, the easiest way to verify is to just try to boot to the backup drive and see if it works. I doubt it will, though.