Do not underestimate those old computers. When you remove audio, visual, and textual data from your programs, remove all the code for handling those and slinging them about and processing them, when you remove the textual file formats and replace them with hand-optimized binary, when you give up on any form of structured modern programming and just hand-compile custom assembler, when your opcodes aren't 64-bits wide, and so on and so on, you can get a lot done in 128KB of code.
Problem was most of the demos used many many megs of ram as they procedurally generated objects and stored them in memory. The 64k demos were disk size.
Assuming memory mapping is used for displays the (128k of ram (64k of words)) is also needed to hold your display buffer. Assuming a mere 320 x 200 x 8 screen will use 1/2 of your entire virtual memory space. Look for demos of what people do on C64 demos for a better idea. Yes you can run a UNIX, yes you can talk TCP... but its going to chew up a lot of the capacity for that infrastructure. I expect very light weight protocols, TCP/IP is likely too heavy weight.
Obligatory (but illustrative of my point): http://www.pbm.com/~lindahl/mel.html