Over 15 years ago I'd occasionally use Net-Tamer:
http://www.nettamer.net/tamer.html
on a Toshiba T-1100 for semi-real things. (Well, okay, connecting to MUDs, a shell account, and the local library's online catalog with the telnet module. Back then you could also still get away with reading news sites on a browser that primitive. These days you're pretty lucky if something is "lynx-friendly" in the slightest.) PPP over a 1200 baud modem in a 8088 floppy-only XT clone works surprisingly well, in that it works at all. So *in theory* at least with just software you can do full-fledged dialup on hardware 30+ years old. (Yes, a T-1100 dates from 1985, but add a 1982-debut Hayes Smartmodem 1200 to an IBM 5150 and you'll have the same capabilities.)
There are some experimental TCP/IP stacks for CP/M floating around as well if you *really* want to get silly. And if we're not drawing the line at microcomputers some of the UNIX versions for the PDP-11 supported TCP/IP, as did TOPS-20 for the PDP-10/DECSYSTEM-20; certainly in principle you could "surf the web" on one of those, though you might have to write your own software to do it meaningfully.
Finally of course there's the "use an ancient computer as a dumb terminal connecting through something smarter" category, which opens the field to anything with a serial port, but I'm drawing the line and calling that cheating. (As much fun as it might be to "surf the web" on an ASR-33 teletypewriter you've clearly gone overboard at that point.)