Networking in DOS (at least for the purposes of transferring large amounts of data back and forth) is really trivial to do. NFS, SMB/CIFS is a little trickier.
Just get a network card with a packet driver (Intel, 3Com and Realtek cards are well supported) and use the ftpsrv program included in the mTCP suite. Use whatever ftp client you want at the other end to upload and download.
Most packet drivers are unloadable, so it's easy to setup a simple batch file to load the driver, start the ftp server and then unload the driver again once you exit the ftp server... so it's not as though it uses any memory when not in use.
For reference, I get around 3-4mbytes/sec out of my amd 5x86 133 and a basic RTL8139C 10/100 pci nic. On my p166 laptop I start up cardsoft dos drivers before loading the packet driver for my pcmcia nic (a basic 10baset one), but still get 1.1-1.2mbytes/sec. Even with only a 3C509B in my 286 based system I could get around 800kbytes/sec.
Much faster than lugging cd's, floppies or zip disks back and forth.
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