To make a private computer network all you need is an Ethernet hub/switch. Plug all your devices into the hub -- no connection to the outside world. (You will have to manually assign IP addresses.) If you want wireless Ethernet you can use a standard WiFi router, just don't attach anything to the "outside world" connection (and, of course, to keep it private, secure the router using strong encryption). (The router should then assign IP addresses for you, but generally won't provide DNS, so you may still want to do manual IP assignment.)
You can also place your server on a standard home network, if you don't want the network totally isolated. Other boxes at home will be able to reach the server, and if your network router is secure then the outside world will not be able to reach the server (unless you want that).
There are other variations where you use two Ethernet adapters per computer, but that gets complicated.
If you want your server to be able to "serve" the outside world it requires a router capable of letting your server be "seen" externally, and your Internet provider may detect the server and require you to get a commercial account (ie, more $$).
1No, that's not how it works. – Raystafarian – 2014-03-29T11:06:31.363
Nobody can track what? Server is just a computer, it still has to communicate through the Internet the same way regular PCs do. (Unless you use avian carriers.)
– gronostaj – 2014-03-29T11:36:07.543Could you clarify what you're trying to accomplish? There are a pretty good number of options to minimize tracking but nothing that will be 100% secure against a determined adversary short of avoiding computers altogether. – Bradley Forney – 2014-03-29T12:24:55.947