You can have a LAN game with GES just like with any other PC Multiplayer game.
Ofc. the setup is also the same: several PCs/laptops connected to a router either via LAN cables or WiFi.
(if you don't know how to set this up, either ask a friend who knows or Google)
Since this is a mod based on the Source engine, you'll also need a own steam account with the free source SDK base 2007 installed on every machine. Using one account on several machines at the same time won't work.
If someone doesn't have a steam account (are there still some out there?), all he/she needs is a valid email adress and the steam client installer (google) to create one for free.
Next there are several possibilities to create a LAN server.
1. one of the clients is hosting a so called listen server through the ingame "create a server" menue
- best possibility for a quick game where the host is taking part himself
- using the most powerfull machine in the room to host the server is recomended
2. create a dedicated server with the srcds
- takes some time for setup, so i'd recomend doing this before the LAN party
- again either host it on the most powerfull machine or dedicate a whole PC just for server hosting
- depending on the setup, the DS doesn't need a own steam account, so it's possible to run the DS and a steam client at the same time.
If you really want the splitscreen experience, you might be able to place the monitors/laptops next to each other and connect every player via controller. (you can use joypads with GES too)