Of course, from a language point of view, the most intensive experience is found by staying with a Russian family. But a home-stay is also invaluable as far as your cultural experience is concerned.
We personally vet all our host families, to ensure both suitable living conditions and a welcoming, friendly atmosphere.
A condition of being a host family is being able to provide you with your own room, so you can get away and have some privacy. But you will share communal spaces, having the chance to relax with your family and share mealtimes.
Food: don't worry, the food situation has improved in Russia, and you won't be sitting down to a diet of kasha and cabbage soup. But it will be different and interesting, as well as copious (Russians set great store by food and will encourage you to enjoy as much as you want).
Host families are instructed to vary the food and find out from you what you do and don't like, without being offended.
As far as possible, all host families live about 45 minutes to an hour from the school (remember, Moscow is a big city and commuting is standard). They are all near a metro station and can come and meet you there the first few days while you orientate.
We encourage families to take you out somewhere at the weekends. Maybe to their dacha or for skiing in winter / picnic in summer.