If I understand correctly you won't actually be playing videos with it you'll just use it as file server (play videos from it using a different machine as video player).
For same purpose (and more) I use oldest N36L hp microserver. It has 2 hard disks in software raid 1, running CentOS (basically redhat linux). The cpu is fine for 100 MB/sec writing or reading which is a lot more than any max quality double blu ray bitrate I might play from it.
I use it as file sharing with samba, transmission for torrents, mpd for audio server (usb connection to dac+hifi), a squid proxy, dns server, web server etc., of course I am the only client for it it's just stuff I play with. The only upgrade I did was switch the 1gb ram to 8 but 1 would be enough for what I use it for.
So for mainly Nas and torrent purposes if I was to look now, I'd try to find a used microserver, any model even oldest. As other possibility I'd check mini-itx.com, I have no business with them or anything but I look at that site when I am playing around with ideas, they have nice case options like small cases, cases with several hard disk slots etc. and obviously mini itx mobos cpus pico psu etc. with all the necessary stuff.
To the question if it wasn't clear, my cpu is AMD Athlon II Neo N36L Dual-Core, it's piss weak but it's good enough for the purpose.
edit: clarifications