I just solved a related problem for myself. I have a previous gen AirPort Extreme, that I'm happy with, as well as a previous gen Time Capsule. The TC is full of backups of old machines that I don't want to throw away. Both devices would accept an external drive, but only over USB2 (to my mind, not fast enough for backups; also, my 2011 MBP has no USB3, so all the cheap USB3 drives were unattractive - preferences for speed were Thunderbolt, GigEther, or FW800 in that order), and I wanted a quick, but inexpensive, way to back up my current MBP (with Time Machine and preferably over WiFi). Would love to get a proper RAID-capable NAS (Synology or QNAP), but can't afford it right now.
After much searching (and considering a lot of alternate arrangements of hardware, ways of attaching "normal" external drives), I ended up getting a 4TB WD "My Cloud" for $160, which is basically one of their standard 4TB external drives, but with a gigabit ethernet jack (and a USB3 port), and enough CPU/firmware so it can represent itself as a network server on your LAN. Supports Time Machine as a first class citizen with no extra software needed (also has optional software to facilitate backups of Windows machines, and does DLNA, iTunes streaming, and a few other bits). With my MBP plugged into Ethernet for the first backup, it did 750ish GB in something under ten hours (wasn't watching too closely), and subsequent backups over WiFi have gone well. And it presents separate shares for Time Machine and "normal" file storage, to keep things tidy. The "Cloud" part of the name is because you can hook it up to WD's website and use software they supply to access it from outside your LAN via OS X and iOS, but I turned off the cloud bit first thing, wanting neither the feature, nor the security liability it presented. The box itself is configurable via any web browser.
Anyway, not trying to sell you on this particular alternative, but I was surprised to find such things available for way less than a traditional NAS.