I find myself in a similar situation -- but I've already purchased a NAS (QNAP TS-639 Pro -- I do not recommend QNAP, btw, my model has had significant durability/reliability issues and even though this has been a widely known issue among QNAP owners, QNAP did not stand behind their product -- but I digress), but the CPU in the QNAP is not capable of transcoding content using Plex Media server. This isn't anyone's fault other than mine, I bought the QNAP prior to really having thought this all the way through.
In my opinion, your decision rests almost entirely upon *how* you intend to stream your media.
If you plan on using an app like Plex Media server to store and transcode your media on the fly, then a "build it yourself" home media server with thin clients at each TV is probably going to be the most cost effective approach. The NAS devices out there that have a CPU big enough to transcode on the fly are almost all in the $1,500+ range (before adding hard drives), while you can build a sufficiently powerful server for less than that. Granted, it won't be as pretty, as small, or as easy, but it will be more capable, more expandable/configurable, more upgrade-able, and cheaper.
If, however, you're going to rip and encode all of your media to a single format (or multiple formats for multiple devices) and won't be doing anything other that "serving up content" to thin clients at each TV then a NAS is do-able. It still won't be cheaper than building your own server, but a NAS will work, and you'll get all of the benefits associated with someone else's R&D, and save a significant amount of time/effort.
How you value your time, and how proficient your are at tinkering with PCs, servers, and NAS software will have a lot to do with what choice you make. If you have time and are wiling to learn, building your own servers and thin clients can be fun. If you want to just install the hard drives, power up, configure your router and "go" . . . a NAS is a quick solution (provided you don't need to transcode on the fly or can afford a NAS with a sufficiently powerful CPU).
The answer to the above questions - what formats of media you have, whether your want to be able to transcode, and time versus budget, etc. -- will drive the big bucket decision of server vs. NAS (from a cost-effectiveness stand point). From there it will be details around what type of server or thin clients you want to build/use or which brand of NAS you want to buy.