In Slovakia you need a Zbrojný preukaz to own firearms (the exceptions being the single/double barrel percussion weapons, original antique firearms manufactured before 1891, flobert weapons (.22 BB/CB cap), and of course blank firing firearms). This is called in English a gun permit. A gun license in Slovakia (Zbrojná licencia) is something different, which is given for business activities around guns-- manufacturing, running a gun store, etc. Unfortunately, to get a license for manufacturing you have to have several years of gun-smithing trade school. Just one of the many professions in Slovakia with protective licensing requirements.
There are several categories of permits. For carrying for self-defence, category A is needed. Other categories include collecting, hunting, and sport-shooting. It is a good idea to join a sport shooting club here. Then you can get a sport-shooting license, which lets you purchase and own any sort of firearm. Likewise, depending on the club, you will get either discounted access to a range or almost free access, and of course the opportunity to participate in sport shooting events-- which range from standing in line shooting at paper targets to the dynamic / combat shooting disciplines.
Up to date information in Slovak can be found on the website of the Slovak gun rights organization Legis Telum.
I have information about getting a gun permit, including a draft English translation of the questions for the gun permit exam.
If you are a foreigner with residency (permanent or temporary) in Slovakia and would like guidance on getting a gun license here, feel free to contact me.