If you build your own NAS I would recommend FreeNAS. It can take a bit of work to figure it out but it is really good once you do. It can do your scheduled backups really well but it can also be a Plex server fairly easily. When I installed Plex on my FreeNAS computer, the plugin didn't work so I had to install it in what is called a "jail". Here is an on the Plex website that tells you how to do it, https://forums.plex.tv/discussion/102532/how-to-install-plex-in-a-freenas-jail .
In terms of CPU power, I have an Intel Core i5 2500 in mine and it streams 1080p fine but so far I've only done those streams to one client at a time. Also if you use the ZFS file system make sure you have plenty of RAM. From what I've been told, for each TB of storage you have, you should have 1 GB of RAM.
Hope this helps