Mac Mini won't start, shuts down on startup chord
Have you added/changed any of the internal hardware at some point?

If your Mac Mini has more than one RAM-block, you could try booting it up with only one block inserted (if it doesn't work with the first block, try with only the second block). That could eliminate faulty RAM as the cause.

The problem is most likely hardware related, and could be for example a defective power supply.