The Internet is not required for you to create a private local area network. To create a local area network, you need at least two network devices, a common protocol (such as TCP/IP) and a device that connects your hosts together, such as a network switch.
When you connects 2 or more PCs connected through HUB/SWITCH/ROUTER, they create a LAN. When you use a HUB all are connected physically. Your OS will have network module and TCP/IP stack will be installed. Configure some IP addresses to those PCs and make sure that they are in same subnet. Then you can avail services like sharing FTP/REMOTE connection all these can possible.
Or, if there are only two computers and you want a very inexpensive option without needing a switch hub, you could just buy a standard RJ45 'Crossover' cable. (If your computers have Gigabit Ethernet adapters see below) This will allow you to network two computers (maximum) directly.
If there are more than two computers, or you are planning to introduce more computers later then a Switch Hub is what you need. Buy a Gigabit capable Switch Hub. (10/100/1000)
If you are setting up systems that you know use Gigabit Ethernet then you need to buy a Gigabit Network cable. If your computers are relatively new, then it won't even need to be a crossover cable either, as the adapters will pick this up. (Check the specifications of your network adapters for this feature: MDI/MDIX. If they have this then you won't need a Crossover Cable, just a standard Gigabit cable will do)