So, I was thinking of upgrading to a funky new 5th generation Time Capsule, with 2 or 3TB of hard disk on-board. It's got exciting new technologies like "beamforming" and 802.11ac. It even looks like a TARDIS (with a white paint job).
But those probably won't make any difference to my existing kit, and the WiFi base station part of my current Time Capsule works fine. So, I thought, why not just hang a 2TB USB HDD of the Raspberry Pi and back-up to that?
Fortunately someone else had already had the same idea and written down how to set up the Raspberry Pi as a remote server for Time Machine, so I just followed that. And it works fine.
Bear in mind that neither the networking or USB on the Raspberry Pi is superfast, but neither was the external HDD I added to it. (Use a disk with a separate power supply). And the backups happen in the background anyway, so speed isn't really that much of an issue.
Note: I have since bought a Synology DiskStation so now I do my back-ups to that rather than using the Raspberry Pi.