Not a NAS but...
I have 3 dumb LCD TVs in the house hooked up to 2 media players, 2 chromecasts and an HTPC. The media players are AIOS boxes and are hardwired into the network.
The HTPC is homebrewed and sits in the basement about 12 feet below the living room TV to which it is connected by HDMI. The media software on the HTPC is MediaPortal, it's free, highly configurable and can do a lot more than most other offerings can. The PC is now relatively old, 7 years, and is a quad core (Q6600) based machine and it now has an SSD boot, I used to use a dual tuner in it but the upgrade to digital TV by my cable provider (spit) means I don't use it as a DVR anymore. I do however have an external USB 3.0 Disk enclosure with 4x4TB drives in it and this is where all my TVs get their content from. 8TB in storage and 8TB of mirrored backup. The content drives are shared over the network and are mapped in the AIOS's, Playon and the HTPC. In the background I run playon server through which I can stream Amazon VOD (I live in the US), Hulu, Netflix etc to my dumb TVs via the AIOS boxes.
Pretty much the HTPC is on 24/7 and this is why I went with an external disk enclosure rather than a NAS. The external disk enclosure (iView HFR2-SU3) replaced a couple of 4 drive backplanes I had in the HTPC case, these were very compact and ran very hot. The disk enclosure is much roomier and the drives run a lot cooler. The iView has some bad reviews around the web but that is for the RAID setup, I use mine simply as a disk enclosure and have not encountered any issues.
The AIOS boxes are now almost obsolete but they are a good indicator of the sort of thing available out there, quite simply I haven't had a legit video file that these things couldn't play, the quality of their output in 1080 actually puts my HTPC to shame.
One thing I have seen with NAS is that the disk read/write speeds on the affordable end of the scale suck! My USB 3.0 connected box will write at about 100 MB/s, makes backing up and copying new BRD images across less painful. My disk setup cost about $650, $150 for each of the drives bought on sale and $50 for the enclosure bought on sale also.