You won't have as much of a chance of falling ill in Azerbaijan as you would in other Asian countries, but a risk is till present. Medical facilities aren't up to snuff throughout the country - you'll have most luck finding one that meets Western standards in the capital city of Baku.
Outside this area, many unsafe and poorly-maintained facilities exist. Sometimes, none exist at all due to the extreme remoteness of a given area, with winter weather conditions and poor roadways causing inhabitants to travel long distances or simply go without health care.
The traveler new to this destination must be aware that there are many very rural villages in this country and the way of life -- and medical system, as a result -- may seem archaic to some. There are issues with limited electricity, infrastructure and other elements that those from developed Western countries may not expect. Things like vaccines and disposable needles are not always available. A 2008 news story casts more grim detail on the issues, reporting that gas, water and electricity are often hard to come by in medical facilities as well.
Staff are not always educated, nor do they consistently show up for work, and there are general staff shortages throughout the healthcare system. You're advised to bring your own prescription medicines on your trip -- with actual documentation that they belong to you, for safe measure --, because you might not be able to find them in local chemists in Azerbaijan.
There are a few scary illnesses possible in this country, such as malaria in rural areas. The stretch from May through October poses the highest risk for contracting the mosquito-borne disease. Tick-borne encephalitis can occur in summer months in this area of the world as well. Other sicknesses include tuberculosis and the H5N1 strain of avian influenza, or bird flu.
You can lessen your chances of contracting flu by avoiding farms and only eating fully-cooked chicken and eggs. It's possible to encounter contaminated local alcohol in bars, restaurants and stores. Drink bottled water, beware of street food and make sure all fruit, vegetables and meat is thoroughly cleaned and cooked before consumption.
There are also a few environmental issues you may come across in Azerbaijan. During the Soviet period, DDT was used on soil in the area, and air and water pollution is high thanks to fuel and petrochemical companies.