I see your 386 and raise you an 8088 at 4.77 MHz and 256K of RAM (or less, I'm not sure what the actual requirements of ircjr is if you want to push it): http://www.brutman.com/mTCP/IRCjr.html

Really, it's just text, it doesn't matter. Literally any general purpose computer will run IRC if it can do TCP/IP and someone bothers to write a client.

Nice. I have an old Cordata 8088 around somewhere at my parents house. I'll have to boot it up and see if I can get a network card for it.

If you do want to pursue that, a couple of tips:

Most ISA network cards are for the 16-bit bus, XT class machines only have 8-bit slots. This doesn't necessarily mean that the cards will not function, but you also need a packet driver and they very often use 186/286 instructions and will not run at all on an 8088.

The Intel EtherExpress 8∕16 will work, so will the 3Com 3C501. You may also have some luck with generic NE1000/NE2000 clones.

Add Gopher to that stack too, with Gopherus.