What is the OS of the Time Capsule ?
The AirPort Time Capsule (the official name, with my emphasis added on AirPort - see below for the reason), runs NetBSD, according to the Examples of use section for NetBSD on Wikipedia. It cites two sources - one being an answer from superuser.com (not very useful in this case), and another an article on wifinetnews.com, that states:
A source unconnected with Apple provided convincing proof that the AirPort Extreme uses NetBSD as its embedded operating system...
The AirPort Extreme Base Station, AirPort Express Base Station and the AirPort Time Capsule all run the same firmware (that's why the AirPort in all the names). This can be seen in the firmware downloads available for these products - it's the same download for all these devices. Of course, nobody has jailbroken these devices yet, so we do not know if or how the installation process chooses specific bits for the underlying hardware. But the OS cannot be different with the same firmware download (it does not make sense for these devices to have such a scheme).
See the recent AirPort update 7.6.4 page for understanding the above correlation.