Get yourself a PCMCIA Compact flash adapter. You can copy your data to the compact flash drive. (requires drivers, so this may fail for you)
You can also get a PCMCIA network card. Probably only 10mbit, but still more than enough. You have Windows 3.11, which was Windows For Workgroups. If TCP/IP wasn't installed on that version, you can download and install WinSock, an early TCP/IP socket stack for Windows. (requires drivers, so this may fail for you)
You can do link between two computers and transfer over serial... but this requires the right software and the right drivers, so this may be out of the question for you.
Emergency Boot Floppy:
Use an emergency boot floppy and boot linux off of the network with a pcmcia network card. You can then mount the filesystem Read-only and copy over the network. (requires linux server, linux skills or someone who can do this for you)
Extract the hard drive directly: (best option)
Yank the hard drive. See what kind it is. Get the appropriate laptop drive adapter cable and relevant USB adapter. Plug that into a newer computer and transfer off the files.
Of the options, the hard disk pull and copy option is really the best way to go, bar none. The only thing you would need to buy would be the right USB-to-laptop hard drive adapter. You can find them on amazon/ebay/etc. If you live near Sunnyvale, California, you can even stop by Weird Stuff and they might have it for a song.