why doesnt your network allow OS older than win2k? having win98 should allow you to connect to the network. its the fastest way for file transfers. you can even use win98 for internet browsing. if you're not really willing to sink any money there(even a dollar or two) then thats really just the only fast way for file transfers. just install the necessary network protocol(TCP/IP)
or use one of your old and slow laplink cables if you have those(serial to serial, parallel to parallel cables). or just spend a few dollars and install a usb card or even just serial to USBport. those things doesnt cost too much nowadays. much better than using floppy or just throwing away that PC.