Try to eat at home and buy food at «green» stores like VkusVill (ВкусВилл in Russian). Avoid city center for rentals. It is quite comfortable to get it by public transport. Cheapest retals are on the north. East is quite dangerous. Khimki — a nearby city on the north is also a good choice for budget in terms of rentals and ecology.
We got information about prices in Moscow on food in cafes and restaurants, products in supermarkets, accommodation rental, real estate and transportation costs based on the data of 172 people.
We analyzed the average prices in Moscow and calculated that one day in city will cost you 6687 Russian rubles.