Bingo. Interlnk.exe and intersvr.exe came with DOS, and as such, are often already on the old computers hard drive. (This is really important if the floppy drive is unreliable.) They can also be copied over the serial lines with a copy command, so on the other end, you can run Windows XP or DosBox under Linux.
Once the file system is under a pseudo-modern O/S, then you can use TCP/IP networking to copy it wherever you like.