Mac mini with a broken HDD, can I use an external to make it work?

Yes. If you're buying a PowerPC Mac Mini as opposed to an Intel one, the drive needs to have a FireWire port.

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Yes but you still have to boot the Mini from either a built-in optical drive or on external optical drive, or an external bootable copy of the OS.

If the computer shipped with Mac OS X 10.6.8 or earlier, use its original disks(phone AppleCare if needed) and an external optical drive.

If the computer shipped with Mac OS X 10.6.2 or earlier, you can also use a retail Mac OS X 10.5(for PowerPC models) or 10.6(for Intel ones) DVD.

If the computer shipped with Mac OS X 10.7 or newer, you don't need an optical drive. Start it up with the Option, Command, and R keys held down; the hard drive may need to be installed inside the computer for this to work.

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First, what Mac Mini is it?

How to identify Mac mini models