The Catweasel used to be the obvious way to put together a more-or-less universal floppy-disk rescue machine, but as far as I know it's no longer being manufactured. Assuming one doesn't have old machines at hand to use, what's the best way to put together a workstation that can rescue data from old floppies of various sizes?
Your best option now is probably the KryoFlux. The KryoFlux attaches to a computer using USB and allows you to sample the raw magnetic flux transitions read by the drive from a disk. The KryoFlux software handles decoding the flux transition data into sector data, but to do so, it requires you to know something about the media you're trying to acquire. Nonetheless, it's a reliable piece of hardware in terms of acquiring a baseline regardless of the underlying encoding or logical format on the media.
However, the license for the KryoFlux for anything other than personal use is somewhat restrictive and expensive. That said, the Software Preservation Society (the organization that developed the boards) have been willing to let libraries, archives, and museums evaluate the KryoFlux before purchase.
The DiscFerret is promising option. Like the KryoFlux, it samples flux transitions from the drive (although at a higher resolution than the KryoFlux), and the hardware and software are both completely open. However, the DiscFerret is not yet in full production, and there are limited numbers of pre-production boards available for sale primarily for developer testing.
If you only need to acquire images of 5.25" floppies, you can use the Device Side Data FC5025, which is relatively inexpensive.
There is a good discussion of this in the Jan. 2012 report, AIMS Born-Digital Collections: An Inter-Institutional Model for Stewardship (PDF). The report discusses a number of hardware strategies for 5.25" floppy disks, including the aforementioned Catweasel and KryoFlux, as well as USB and motherboard solutions. There are also discussions of other tools such as AccessData FTK (Forensic ToolKit), Curator's Workbench, FRED (Forensic Recovery Evidence Device), etc.