Older PDP-11's with 18 bit physical addressing (max 248 KiB of memory) certainly could support a few users on serial terminals.
To be fair, programs were loaded from disk and swapped out to disk, when the memory became full. Programs also used disk based overlay structures, so that each individual task could fit within the 64 KiB address space. Adding DecNet and the Ethernet card(s) further reduced the amount of memory available for applications.
Also truly 8 bit systems, such as Intellecs (8080) or Excorcisers (6800) with max. 64 KiB of memory heavily relied on loadable programs and program overlays (at last the compilers).
The OP wanted a TCP/IP stack, which requires some buffering for realistic performance. A bit mapped LCD even with black and white colours only may require more than 10 KiB of memory for the bit map. A few KiB is required for the character generator tables (these tables were in the character generator ROMs on VDUs or the character forms were integrated into the printer daisy wheel).
An 8 bit processor is typically only capable to easily access up to 64 KiB of memory, so fitting a stand-alone system is quite a challenge.
Of course, if the TCP/IP requirement is dropped and only raw Ethernet or UDP frames are used on the network and a PC etc. with some disk space, is used as a virtual mass storage device, the 64 KiB would be enough. Loading programs or overlay segments from the PC file server using UDP requests could be a nice replacement for floppy disks in old systems :-).
Paul