I have two Macs at home which use Time Machine to back up to a shared Time Capsule. Does anybody know of any tools or methods for other operating systems to perhaps make use of that back-up facility? That is, does anybody know of a Windows or Linux client for a Time Capsule? (Even if it's a very limited one.)
Super highly unlikely you'll find something like Time Machine, as that relies on file-system event logging that was custom made by Apple for Mac OS X. You can, however, mount the Time Capsule's disk drive as a file share using either AFP or SMB and use any program that will back up files to a file share, which is almost any program that backs up to disk, although beware of possible issues relating to ownership and permissions and unflushed buffers.
Note that the following setup assumes you are using Airport Utility 5.6 or later but NOT Airport Utility 6.0. If you are running Lion, 6.0 is installed by default but you can have both installed on your Mac and use both. You can download Airport Utility 5.6 from Apple. If your are on Snow Leopard or earlier, you probably already have 5.6. Airport Utility 6.0 is greatly simplified and missing a ton of features that were available in 5.6.
You need to explicitly enable File Sharing on the Time Capsule using the AirPort Utility. From the AirPort Utility, select the Time Capsule, select Manual Setup, click on the Disks tab. That should present you with the "Disks" pane, which should list the disks in and attached to the Time Capsule. Note that like Disk Utility, the list is hierarchical, with drive partitions listed under drive devices. My Time Capsule just has the internal drive with one partition. (Click on the triangle next to "Time Capsule Disk" if you don't see a partition name under it.) Click on the partition name an the right panel changes to show you information about it, including giving you the ability to change its name. This partition name is the root directory name for the SMB share, so make it short and without spaces. You should avoid hyphens, too, as sometimes hyphens are converted to spaces and sometimes spaces are converted to hyphens and it just gets unnecessarily confusing. Underscores are OK.
Now select the File Sharing panel to get to the rest of the configurations you need to set.
With Time Capsule firmware version 7.6 and later you have 3 choices for password protection: you can use the same password for file sharing that you use fo Time Capsule, you can use a different password, or you can go all out and set up multiple users, each with their own password (and corresponding ownership and permissions features).
You also want to assign it to a Workgroup if you're using SMB and you want your Linux and Windows machine and the Time Capsule to all be in the same workgroup. The default workgroup name of "WORKGROUP" is the easiest way to go.
To authenticate the SMB client for access to the Time Capsule when you have not set up individual users on the Time Capsule, you can use any (non-blank) username (it is ignored) as long as the password is correct.
I would recommend against even trying. I did this myself with a Windows computer a couple of years ago using some undocumented features of Time Capsule that I figured out myself. Then I needed to restore a Mac from my Time Capsule backup and I discovered that I had interfered with the Time Capsule's ability to work with the Mac. I created my own tech-support nightmare. I spent hours on the phone with Apple and got to a senior tech support manager who said, "That's why what you did is undocumented. The Time Capsule is not designed to support backing up from Windows computers. We don't encourage anybody to try this. You found out the hard way."