I guess what I'm trying to distinguish is the difference between a server, and just sharing a folder on our computers. I'd like it to be able to stream media if we throw it together. I have a ps4/xbox360/smart tvs all over the house. Would it increase performance by having my downloads go to the server over coming to this computer?
That depends on what your machines specs are like.
A box used just for dedicated storage will perform better than a machine that is being used for other purposes as well, just like an i7 will perform better than a Core2Duo. If there's more resources, there's more things that can be done at the same time.
Then there's network speeds. Is it going to be wireless? are the devices accessing it wireless? Multiple wireless links will destroy speeds, and reliability.
I personally run a scale out file server at home, for my Hyper V vms. This creates a shared storage for failover, but is very limited in other things it can do. This is a function provided by a server OS, wheras some things don't work if you just create a file share on a consumer OS.
For media streaming , a dedicated hard drive for storage on a machine directly connected to the router should serve your purposes fine.
It all depends on what you're doing with it, as previously stated.