This beautiful courtyard hotel in Kashan is located in the old town and has basic but nice rooms. Some of them are equipped with bathrooms and AC, others have only a fan and a shared bathroom. We stayed one night in a room with AC and bath and one night in a room with a fan. However, since the room with the fan was downstairs, the temperature was in fact much better than in the room with AC. Both rooms were simple but nice and very clean.
The very best thing about the place, however, are not the rooms, but the general atmosphere and the staff. While the hotel is by no means a backpacker place (possibly being at 15-25 euros a room a little too expensive for that), but still the atmosphere in the courtyard was very social. The hotel has a traditional seating courtyard restaurant / tea house (but you can bring your own food if you're a guest) that encouraged interacting with other guests. The courtyard restaurant serves great food, although sometimes a little slow (1+ hour for two meals was not uncommon). In addition, it was one of the nicer courtyards we saw in Iran: very green and well tended.
The staff members were all extremely polite and helpful, assisting with planning the remainder of our trip to Iran.
Minor point of advice: the staff tends to recommend trips to the dessert with local guides, to be booked through them. While prices are fair, the trip might not be worth it if you've been in the desert before or planning on visiting it somewhere else.
Do not miss Fin Garden if you're in town!