I've used Synology NASes and they are great but they really are single purpose appliances. If I had one, I'd need a server too for my use.
For example, I run a Plex server and it can end up transcoding 3-4 1080p streams at once if everyone in the house is using it. Not sure there are any NASes about that can handle that, but the i5 4460 I have in it is fine.
I run VMs with Linux webservers etc in, and a few other things, so it has 24GB ram.
The host OS is actually Server 2012r2 currently, only because I wanted to play with storage spaces and ReFS, I had some spare SSDs so I have a tiered storage pool.
Yeah, I'm going outside the normal usage profile, and in my case, it is bulkier than some - it's in a Node 804 case - but it can hold ~10 HDDs + 2 SSDs.
I could have done it in a small minitx case like this
http://www.chyangfun.com/cfi-a6039.html with an ITX board if I had been tight on space (it's in my cellar) and wanted to spend money (I built it from my spares cupboard other than the case, even the HDDs I salvaged from spares)
The whole thing cost me £69 for a case, and that's it.