We stayed with a Mongolian family who let out two ger huts to tourists. It was organised through Happy Camel Expeditions, who operate from the 'Chez Bernard' cafe in Ulan Baatar.
The accommodation was more than we had hoped for when we pictured a ger. We had the hut to ourselves: two beds, a dresser, fireplace, and sink. It was surprisingly spacious, in a tardis kind of way, and much more comfortable than you might imagine.
Facilities were obviously limited. The "sink's" water supply was just a small tub with a tap sticking out, and the toilets were of the squatting variety in a small shed twenty metres from camp. The primary washing facility was a stream 15 minutes' horse ride away.
But its appeal doesn't come from its amenities. What made it so great was that you were detached from the world, surrounded by incredible scenery in a very relaxing environment.
The food was plain, but it was brought to us in our ger like room service, and it was absolute bliss after a day on horseback.
In terms of accommodation, there are far more luxurious places to stay, but in terms of experience, it is hard to beat. To anyone visiting Mongolia, I would definitely recommend staying out in a ger (I would recommend Happy Camels too, as some other gers around the park looked incredibly synthetic and touristy).