For a 68K on-computer solution, I think you'll need to explore the path that
@s_pupp mentioned. Use a tool (maybe FEdit) or a low-level disk copy tool like Copy II Mac to read every sector on the disk. One or more sectors will probably fail to read, but with some manual effort you may be able to piece together the rest, especially if it's text documents. Here's what you can probably expect to find on an 800K floppy:
sectors 0 and 1: Boot info, you can ignore this
sector 2: The master boot record, there should be a duplicate copy of this at sector 1598 (decimal). You can probably resconstruct this if necessary.
sector 3: The volume bitmap, each bit indicates whether a sector is used or unused. For read-only use, I don't think this info matters.
sectors 4-15 decimal: The extents overflow B-tree. This probably isn't used, unless the files on your floppy disk are heavily fragmented.
sectors 16-27 decimal: The catalog B-tree. This stores the info about what files are on the disk, and their names, locations, and sizes. If these sectors are bad, then you're in for a rough time.
sectors 28-1597: file data
sector 1598: duplicate MBR
sector 1599: unused
The locations of the volume bitmap, extents overflow file, and catalog file can theoretically be different, and all are specified in the MBR. But this seems to be where the Mac's normal 800K floppy format routine puts them.
If you can make an image of the disk, I can look at it and see what I can figure out. I recently made some Python scripts to analyze disk images at this level and attempt to extract files.