It's probably going to cost you time, more than hardware, I'd honestly suspect? Any simple Atom-based NAS should really have enough "oomph" for like, a REST HTTP API that you can talk with. I have a 4x core C2000 about a foot from me that would do great. You can piece one together for a cheap few hundred $ (like $300 can probably get you something decent). But you're going to have to program all the shit together, that's consuming.

I've been looking at this for my home. Realistically I've just been spending a lot of time so far just modeling my database schemas so I can eventually put APIs on top of it all to talk with.

Mostly just make sure you pick some devices you can think you can interface with, have been reverse engineered, or get ready to reverse it yourself I suppose (which might be easier than you think, considering how bad embedded security is)...