I was going to say that a mac mini is probably more than you need. If budget is a concern, and you have the technical skills, an Intel NUC or Zotac Zbox would probably come in a few hundred dollars cheaper and work for you.
You would need to deal with either Linux or FreeBSD (or pay for windows) to go the NUC/Zbox route. So, this really depends on your ability (and patience).
I'm currently running Plex on an Intel NUC with an i3. The media library is stored on a pair of USB attached hard drives (Seagate 5TB) that are mirrored. It can transcode full 1080p without breaking a sweat. Point being, even a used Mac Mini with USB drives for media should work.