Maybe I've been around big business industry too long, for me a server is something that's plugged into the internet 24x7 running applications. The OP did not say anything about running a personal server for a limited number of personal devices. I can see, in this case, where a nice cubby-hole for a desktop PC or something with similar capabilities might be useful. Basically, all you're wanting to have available is something that will allow you to share data between personal devices. That I can understand.
When my son was in college he took computer programming classes, he also worked at Best Buy on the Geek squad. He built himself a "server" and was part of a gaming club that was in the process of coding their own software computer game. Each person contributed code. It was several years in the making, and I don't know if it ever really got finished. But one thing he had to do was keep his computer up and running 24x7, connected and his computer was a "server".
On a small private scale, off the internet grid, I see where your idea might be beneficial. But I would think a simple lap-top would serve the same function if all you're trying to do is share data between a few select devices.