Forget all I said about USB....
What you could try... dig up your floppys and try to boot from them. If that works, get out of the setup (hit the F3 key) and from the command prompt (make sure you're at the root of the floppy) try:
- SYS C:
- copy CONFIG.SYS C:
- copy AUTOEXEC.BAT C:
- FDISK /MBR
The first command adds the needed system files to your drive, you then copy the two startup files to the C: drive and the FDISK command should sort out the master boot record for your C: drive.
I'm digging deep in my memory for this and have looked up some stuff online so cannot guarantee anything. Removing the drive and using this in an external caddy is still the safest way to get to your data if you ask me, but if you're up for an experiment and willing to take the risk, have a go!
Edit: You used to be able to install Windows 3.1 over an existing installation, keeping your files intact, but if you can't boot from the drive, that option will not be there I think.