Sorry for being confusing.
NAS stands for Network Attached Storage. It is basically a hard drive stuck on your local network. They are primarily used to store videos and pictures so they can be read from things like DVRs and streaming media players like appleTV and Roku players. They can be pricey, in the 140$ range, but they make great dedicated server appliances. (It is what they are designed for.)
Under the hood, they run a very minimal linux kernel, busybox, and some other server software to help them do what they are designed for. Basically they are tiny linux boxes, that dont look like they are tiny linux boxes. You dont need to be a linux geek to use one.
With some ... assistance... they can be made to run OTHER kinds of server software. Like the minecraft server software. ;P (but it DOES take a linux geek to make that happen.)
Once it is running on them, you just connect to the server's IP address like any other, and it should just work. The only real issue might be getting past your router-- you might need to set up a forwarding rule on the router to forward the ports the server uses so that people can connect to it. That's about it. You would have to do that if you used one of your own computers as a server as well.
The black magic voodoo incantations are only for people like me (there are some in this board), who might be interested in trying this kind of thing themselves.
If you want to hire out the server, that's fine. I would just set up the server on something like my NAS, but what is easy for me is not easy for other people. Paid servers exist for a reason.
I fully intend to continue my experiment though. I have tomorrow off, and will use it to finish the experiment. My NAS is always turned on, but again, my upload speeds are horrible. You are welcome to help test it though.