This may be a silly question, but why get a whole extra machine just to make 4tb available on the network? I have an old Linksys router with a USB port that allows you add USB storage to your network, that may be a good place to look if your needs are basic.
Additionally, it looks like you are going for a relatively powerful machine to be able to access a minimal amount of data at a slow speed. Have you considered just getting a couple of external hard drives and just hooking them up to a Raspberry Pi?
Going the Raspberry Pi route would cost $288-$333 depending on accessories. This could be a really good route depending on what all you want to do.
Another way to save more would be to just add the drives to an existing desktop, and just setup samba/nfs to run on it. You’d just need to leave the desktop on all the time, or turn it on when you need to access it elsewhere.
Also, for reference, I have a NAS with 40tb of storage that I use as a VPS host. I have several virtual machines running 24/7, including a Plex server. I stream many videos locally, and have many users that access my content regularly. From what I’ve learned over the years of running that machine, I know that as long as you are just directly streaming the files, and not trying to transcode them, you don’t need hardly any processing power. I don’t have a GPU in my server, and don’t see a need to put one in. If your needs are really as simple as your post suggests, the raspberry pi route would be the way to go. It would also allow you to dip your toes into running a NAS, and see where your original build was lacking, and give you a better idea for what you want your next evolution to look like.