This is a beautiful little bed+breakfast run by an equally beautiful family. It's located in the tiny village of Ayvali, away from more touristy towns which I recommend avoiding if possible. Without actually staying in one of the local homes, this is as "authentic" as you could want. The owners have built 3 rooms and a small dinning area abutting their personal home. The wife cooks delicious, large, traditional meals for breakfast and dinner. The son speaks excellent english (his parents do not appear to speak any) and can give you tips as to where to visit, depending on time constraints. The local men gather at the town cafe in the evenings to relax and drink tea. There are also some cave dwellings to be seen in the village, though they pale in comparison to others to be found in the region. All in all, it really is a wonderful experience. Highly recommended.
Two warnings:
First, Cappadocia is a region, not a town, so you will need a car to get around. Many people visit with a tour group. I prefer more independent travel, so I rented a car in Ankara and drove, which was a great experience.
Second, Ayvali is a tiny little village and I had some trouble locating it (used Google maps). I would recommend avoiding the route that takes you through Mustafapasa, and instead take the route that goes around it.