If you are planning to spend your time in the outdoors, away from medical facilities, it is a good idea to bring your own emergency kit. Emergency evacuation insurance is also recommended. There are a number of private clinics which provide more comfortable facilities than state-run clinics. If you need to see a doctor for an infection or minor problem, these private clinics can be immensely more pleasant. However, their existence is still in the initial stages.
New clinics disappear as quickly as they emerge.
If you have a real emergency, do not spend time searching for a private clinic. Find an interpreter and have them call an ambulance or get yourself to a hospital. To be safe, memorize a few Russian phrases before you leave, just in case. If possible, have someone who speaks Russian accompany you whether your ailment is minor or severe. As a general rule, most Russian doctors speak very little English or any other foreign language.