Network cards always have widespread driver support, it's important enough,that it works. There is always a DOS driver even - it's small and simple, and there are all sorts of deployments and other tools that use DOS.
Still to run a 3COM board on a Windows 7 32bit PC, I had to download an old driver for Windows 2000 (or XP and 2000)
If you try linux there should be driver support for the old SCSI card. Old stuff tends to be supported forever (but these days still with a combined printer + scanner device, the printer might work but not the scanner). Old stuff that's too rare so no kernel developer/maintainer has access to it and/or where there is very little interest is eventually dropped.
It should be at least possible to do a raw dump of the whole disks.
Else I have the same suggestion as orangpelupa, but do it from DOS maybe.
It's possible to run MS-DOS 7.10 from a USB stick, and that version supports fat32. Although it is likely that you could make a 1000MB fat16 partition and run DOS 6.22 or a crazy old one without problems.
If that's an Amiga file system on your old hard drives there's likely old freeware tools for DOS that allow to read the files.