I guess on Linux you can get with one computer, 4 wine prefixes, so 4 fake Windows setup for 4 GE:S installation, then 4 windowed noborder games running on same screen (setting arbitrary sizes to half the height and width and manually placing them, or use a tiled window manager). There is an input issue (1 keyboard and 1 mouse shared for all the instances running) but if you use joysticks it's probably not a problem, using one for each instances. Then you have the network issue, you probably have to setup a virtual network bridge and 4 virtual network interfaces, then run each instance with a different ip address. In fact it means you don't need a physical network at all, that is good. Then, there is a question: is steam allows to run multiple instances of the same game? You can run steam 4 times on the same computer due to the 4 wine prefixes (it's just lke 4 computers running one windows each), but if I remember right, Steam prevents you to run a game you are already playing on another computer (or will disconnect the game on the other computer). So perhaps you need 4 steam accounts.
So, if you have a computer with an AMD GPU (because of
that), Linux skills, wine skills, knowing how to install unreleased drivers, unreleased wine, unreleased gallium-nine, chroot skills, X11 skills, virtual networking and bridging skills, hl2 switch knowledge, 4 joysticks (and perhaps 4 steam accounts), yes, probably you can do a GE:S 4 player splitscreen with one computer and one display. But it's not easy.
Btw, I haven't tried joysticks on GE:S/Wine/Linux.