I have a base model Mac Mini for an iTunes server, it just runs iTunes with home sharing 24/7 and all my movies are on a fast USB 3.0 drive. You could plug multiple drives into a Mini (or any other computer running iTunes) also, but that gets a little more complicated. My setup works really well with my two Apple TV 3's and a MacBook Air. I also have a screen connected to the Mini, and it's connected to my stereo system so I can also watch iTunes movies there.
I have a 2TB Time capsule and played around with it a bit for this. External drive performance is awful, the interface is USB 2.0, expect 20 to 30 MB/sec speeds. This is fast enough to stream video, but there is a lot of latency, which makes it feel sluggish. Also, I believe the disks will automatically spin down if you stop using them for just a brief period of time. That also makes it feel really sluggish. Using a dedicated iTunes server is a much better user experience IMO (however I only have an Apple TV3 so I never tried any of the apps available on the Apple TV 4).
Backing up your media is also a pain using drives attached to a Time Capsule. Time Machine won't do this. So you have to connect an external drive to a computer on the network and back them up as shared drives. This works, but can be painfully slow with a lot of media, since the bottleneck is the 20-30MB/sec throughput on the disks. My library was about 1TB at the time I was experimenting with this, and it took hours to do a backup (using direct wired gigabit ethernet).
With the dediated machine for an iTunes server, you could just use Time Machine. I use Carbon Copy however, and every night my media drive is cloned to an identical USB 3.0 drive that is also connected to the server. I rotate between two backup drives and keep one of them separate.
If you have a big investment in media, you should also have a backup strategy.