There are many exchange places, both in airports and in city proper. Technically and officially they all are bank branches, but often it consists of an exchange office only.
The airport exchange rates are not the best, though if you are not changing lots of money, probably the price difference will not be worth your time.
You can withdraw rubles from ATMs (and certainly you can pay by card, which is widely possible; in fact, you will not need much cash), unless your bank prohibits this; check with your bank in advance. Also find out the fees they could charge and the exchange rate they would apply; you may find the exchange rate quite decent. (Only beware that exchange rate changes daily or even hourly, so compare exchange rate of bank with current Moscow exchange rate.)
In fact, I don't know about French card in Russia, but in opposite situation (Russian card in Europe), I often find it to be the best rate to simply pay by card wherever possible, and withdraw from ATM if I need cash.