Movie Library using Lacie 4 tb auxiliary drive

Hi, I am new to Apple and recently discovered that I can view my movie library that I have burned to my MacBook Pro and can view it on my Apple TV. I want to be able to watch my whole library, in DVDMEDIA, but I noticed that I have to keep my laptop on for it to work. Can I use a mac mini and put a 4 tb Lacie Thunderbolt drive connected to it and watch movies without having to access the drive with the computer everytime I want to watch a movie? Or will I have to choose the auxiliary drive on the mini to open the drive to access the movie?

Any help would be appreciated!🙂