In theory, most foreigners are entitled to free emergency care in Russian hospitals, and some countries have reciprocal agreements with Russia for free healthcare for their citizens. Russians are extremely proud of the quality of their doctors, but few can deny the chronic under-funding of the public healthcare system, and free treatment is almost unheard of even for Russian citizens - even if you are admitted to hospital free, you will almost certainly have to pay for medication, and for reasonable treatment by nurses. It is therefore well worth taking out effective travel insurance.
There are a large number of private clinics in Moscow, many of them with English-speaking staff and many with their own ambulances. However, there is not necessarily any relationship between cost and quality, and the best way to ensure good treatment is by recommendation. In a real emergency, you should be able to organize a medical evacuation in one of those private ambulances.
Booster-shots for diphtheria, polio and tetanus are recommended for travelers to Moscow. Drinking water straight from the tap is not advisable, although you'll be hard pressed to avoid eating food that has been washed in it, and Moscow's water is not known to contain any specific evils.
The emergency services number for Russia is 01. For an ambulance, say: "skoraya pomosh", literally meaning quick help.