The Retro PC Solution
One if the most obvious solutions is to use an old PC from the era that has an old floppy drive. This is not necessarily the best solution. First, you must have to have the extra space for the old PC. If you have 3.5" and 5.25" diskettes to archive you need to have both 3.5" and 5.25" drives to read them. Disks for other platforms often are not readable with a PC drive and controller. Even if you do setup an ancient PC to access the data on the diskettes, you need to have some way to get the data out of the old PC. If it is recent enough to have USB or has a PCI bus for an aftermarket USB controller you can use a USB thumb drive. You can add a network interface controller to connect to your network or broadband modem. You can also add a CD burner. With these solutions you will need to track down older versions of the support software, such as a burning program.
External USB Drives
A more tenable solution for most is to use an external USB floppy drive. These are affordable and readily available, but be sure to get a drive that can read both DD (double-density) and HD (high-density) disks.
External 3.5" USB Drive
The FC5025 USB to Floppy Controller
The Problem of Uncopyable Data
However, there are some cases where simply copying the files is not a solution. It is often important to preserve other aspects of the original media, such as directory structure, disk labels, etc. Disk images can usually capture all of the extra information, thus is a better solution for archival purposes. Identical disk copies can be made from the images. While there are several programs that can do this, many of the standard programs are now dated DOS only programs, such as RawRead/RawWrite, TeleDisk, the MS Diskcopy and several others. The best known modern solution with a GUI for this task is WinImage.
Another problem we'll encounter is the preservation of disks from other platforms that cannot be read natively in modern Windows. The lack of a proper driver to read platform specific formats or even hardware differences can make it difficult or nearly impossible to do so. Apple, Atari, Amiga Commodore 64 and Mac disks require additional measures to access the data.
Then we must take into account that not all data cannot be easily imaged. Some copy protections schemes of the floppy era present extra hurdles. The Copy Protection Control (CPC) that Sierra used for several of their AGI game releases utilized a fake bad track to hide part of the copy protection that the interpreter would check to obtain the decryption key pass to the executable. Without this track, the game will not start. Since the OS saw this as a bad track, it would ignore it and could not copy it, either as raw files or as part of a disk image. Dealing with some of these disks could be dealt with using special hardware, such as the old COPY II-PC option board, using an OS that allows low level access. For more on the CPC protection scheme, see this post.
The Catweasel 4
There are modern solutions for imaging these problematic disks. There was the Catweasel, which many considered not completely satisfactory and seems to now be defunct. The KryoFlux from the Software Preservation Society provides a floppy controller that can interface with a modern computer (Windows, Mac and Linux) via USB.
The KryoFlux Controller
The DiscFerret Controller