Many families have opened up their homes to foreign visitors as "bed and breakfasts". Some B&Bs services can arrange the "visa" invitation needed for obtaining a visa, provide you with a private room in an apartment, or sometimes an entire fully furnished apartment, and provide as many meals as you want. In some cases they can guide you around the city, arrange for tickets, translators and meetings with people in business and government.
There are many organizations in foreign countries as well as in Moscow that can arrange such home stays. In addition to home stays, there are agencies that provide short term apartment rentals with various levels of service.
Sometimes at the railroad station you can find people offering a room for overnight stays but be cautious when pursuing these options.
Rather, call the people who advertise short-term flat rentals in The Mowcow Times. For rentals of three months or less, see Organizations Offering Short-Term Flat Rentals in Moscow. For other budget accommodations or rentals geared towards students see list of cheap hotels given in HOTELS.