Cappadocia has a great number of such subterranean system, some people say 200 although more realistic estimates point to around 40, with many still not discovered or excavated. The largest are Derinkuyu and Kaymakli which are believed to date back to Hittite times. As time passed, the tunnels have been extended and ultimately served as shelter for thousands of Christians.
The Hittites were an evolved bronze age civilization which was among the first to use iron and reached its peak around the 14th century BC. Around 1180 BC the empire disintegrated and various "neo-Hittite" states were formed under Assyrian rule, sometimes engaged in wars with one another, this until ~700BC. The Persian conquest followed between the 6th and 4th centuries and then Alexander the Great and the spread of Hellenism in the 4th century. Apart from Xenophon I don't know any other ancient text describing underground homes in central Turkey. I also want to mention that the Hittites had an Indo-European language and it seems that they used the Akkadian cuneiform writing system. I don't know how Hittite texts are being translated but it seems strange that these underground systems have not been mentioned in any of them.
Recent studies have concluded that the Phrygians, a people with a language much more related to Greek than to Anatolian languages arrived in the peninsula around the time of the downfall of the Hittite Empire and so the underground cities, or a part of them, can also be attributed to this civilization. Some researchers even say that these underground cities are described in the Vendidad, in Fargard 2 (click here to read). I however am reluctant to embrace this opinion not because it couldn't be true, but because more research must be made towards this direction.
I will now show you some pictures from a couple of these underground cities so you can imagine how life was in them.
For more info:
Countdown to opening of Cappadocia's second underground city [Saint Mercurius]
arhZONA - Prodzemni Gradovi
Ancient Standard: Derinkuyu - Turkey's Underground City