Apartments can be rented in Moscow for longer stays. Prices for foreigners range from a very cheap $100 per month on the outskirts, to an outrageous $2000 per month for a 3 room apartment in the center. It all depends on connections, currency, location and negotiating ability.
Staying in a private apartment legally is still tricky from the viewpoint of old regulations and should be clarified. For example, prolongation of a visa often requires that your passport be registered with the "Registration Office" OVIR, and this in turn requires permission to stay in other than an official hotel or as an invited guest in someone's home. See VISA OFFICES AND REGISTRATION. If you want to rent an apartment and observe legal regulations, you need to have a written contract with the proprietor or REAL ESTATE AGENTS. Many apartments, however, are rented informally on the basis of a handshake, even those which are offered to you by apartment rental services. In some cases rent is paid in cash. Occasionally you can pay by bank transfer and the price should include NDS (value added tax).
Rooms can be found with many families who will free up their own room or more likely a grown child’s room for a very reasonable rate. There are a growing number of real BED & BREAKFASTS that can offer you a room of your own and meals.
Hint: Look for apartments with a gas or electric hot water heater. Find out about trash disposal, cable TV, where to pay your phone bill and how to get communal repairs done. Ask if there is someone who can clean, do your laundry, cook and shop for you. This can save hours.