It is indeed relatively easy to get medical help in Costa Rica if you're a traveller.
By almost any standard, Costa Rica has some of the best healthcare in Latin America. There are two systems, both of which foreigners can access: the government-run universal healthcare system, Caja Costarricense de Seguro Social, known as Caja, and the private system. Both health systems are constantly being upgraded – new hospitals, new equipment, and improvements in staff training.
Private: There are three private hospitals in the San Jose area: CIMA, Clinica Biblica and Hospital La Catolica. Clinica Biblica is part of the "Blue Cross" network so if you have Blue Cross coverage you may be able to be treated there under your insurance plan. Of course, you would need to check the specifics. All three of these hospitals are staffed by English-speaking staff and can provide the highest level of medical care. But it's not cheap.
Both Clinica Biblica and CIMA now have facilities near Liberia, not far from LIR airport. Also in Liberia, there is the private hospital San Rafael Arcangel which has an excellent reputation and quite a few of their staff speak English.
Public: Costa Rica has a comprehensive system of public clinics and hospitals – formally known as CCSS – and tourists can use these in case of emergency. The cost is low, if anything. The public clinics are not equipped to handle sophisticated emergencies and will usually just try to stabilize the patient and send him/her on to a hospital. Some hospitals in remote areas are not fully equipped to handle serious emergencies.
Ambulance service is available, operated by the Red Cross and is free, although many of them designated as "Basic Support" are little more than taxis with a bed. If you had a severe accident and your life is at risk, call 911.
Like doctor's offices and clinics, pharmacies are everywhere and there are quite a few drugs or things like asthma inhalers that can be bought without a prescription. All pharmacies have a relationship with a doctor (or two) and can refer you to the doctor if you need a prescription. Some larger pharmacies have doctors that are at the pharmacy all the time.
Trip Insurance: There is a wide array of trip insurance available and you should look carefully at the "fine print." Some require you to pay up front and then put in a claim when you return home. Others take care of any costs. Pick the one that works for you. Generally, World Nomads is a good company you can purchuse your travel insurance from.