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.
Before you travel
For the reasons stated above, it is well worth taking out effective travel insurance for your trip to St. Petersburg, or ensuring that your current travel insurance covers the city (you may want to know exactly what services they cover, as well).
Booster-shots for diphtheria, polio and tetanus are recommended for travelers.
While in St. Petersburg
If you do have health problems in St. Petersburg, there are a large number of private clinics in the city, many of them with English-speaking staff and many with their own ambulances. In fact, Russia is becoming an increasingly popular destination for medical tourism, as many procedures are available here at much lower prices than in the US or some parts of Europe. In a real emergency, you should be able to organize a medical evacuation to Finland.
The emergency services number for the ambulance service is 03. For an ambulance, say: "skoraya pomosh", literally meaning quick help.
A few factors to be aware of during your stay...
The situation with tap water in St. Petersburg has recently improved, with a new purifying plant in the city bringing the water supply up to international standards of safety, at least at source. Unfortunately, the pipe system in many districts and buildings is very old and neglected, and by the time the water comes out of the tap it often has quite high levels of heavy metals - and may be a rather unpleasant yellow color. Therefore it's still probably safer to drink bottled water, unless there's a filter system installed where you're staying.
Every winter when there is a reasonable level of snowfall followed by a freeze, large quantities of ice form on the roofs of the city's buildings. When temperatures rise above freezing, this has to be cleared quickly or it is likely to break off in large chunks and fall to the street. Unfortunately, the municipal authorities seem unable to organize this process effectively, and there are regular casualties among pedestrians. There is not that much that can be done to avoid randomly falling blocks of ice, but it's worth keeping your wits about you during sudden thaws, and keeping your time on the street to a minimum.
This is more of an irritation than a real health risk, but by Western European standards the number of mosquitoes during the summer in St. Petersburg can be very high, particularly anywhere near bodies of water (which covers most of the city). If your accommodation does not have netted windows, you will probably want to invest in a chemical anti-mosquito plug-in.
Available in all supermarkets, pharmacies, and many kiosks (Raid and Raptor are popular brands), these simple devices will only cost a few dollars with refills (either tablets or screw-in bottles of liquid). Plug the unit in a couple of hours before you go to bed and you should be able to sleep peacefully. If you are planning a trip into the countryside, you will definitely need to buy some kind of repellent cream or gel, also widely available in supermarkets and pharmacies.
In recent years, the risk of tick-borne encephalitis has spread from its nexus in Western Siberia all across Russia and most of Europe. There have been (very few) incidences of infection in the countryside around St. Petersburg, and local ticks are also known to be carriers of Lyme disease. There is no risk in urban areas, and this should only ever be an issue if you are involved in outdoor activities (camping, hiking, fishing, etc.) in forest areas. If you are, it is worth keeping as much of your body covered as possible (this will also help against mosquitoes).
Should you have the misfortune to be bitten by a tick, the most important thing is to have it removed quickly, usually by rubbing the area around the bite with alcohol or cooking oil and then using tweezers to grasp the tick as close to the skin as possible and pull gently but steadily until the tick becomes disengaged from the skin. It is vital that you keep the tick that bit you for analysis, and this is best done by placing it in a sealed container with some moisture - a recently emptied water-bottle is ideal. Provided the tick is promptly removed, the risk of infection is very low, but you should still contact a medical professional as soon as you get back to the city.