A family doctor won't accept you if you're not insured.
There might be a case that you know that you should be insured and the system shows you aren't. Then you have to sign a special paper that you claim it's a mistake and you have to explain it with your employer who is in most cases responsible for your insurance.
The previous system was such that every employee had a special booklet where the employer had to stamp regularly that the health insurance has been paid by him to the ZUS. And you had to take this booklet with you when you went to the doctor. Then they were going to start this computer system so they stopped issuing new booklets, but it turned out the start of the system had a delay of a few years. So if someone had no more free place for stamps in the booklet, he had to show at the doctor a document from the employer confirming that the employer has paid the insurance (which looks so:
formularze.iform.pl/zdjecia/formularze/6471/1329589097raport-miesieczny-dla-osoby-ubezpieczonej.png
- and shows also the specific amounts of money paid, so it betrays how much the patient earns, so it might not be comfortable, but it was a temporary solution).
As the people before me said, if you are insured in the national healthcare system (which is done by the employer, employer of a family member, in some minor cases by a university, or you can pay the insurance or your own), you don't pay absolutely nothing for anything that is covered by the healthcare system. Unless something is paid by the healthcare system only partially, of course (it's often so with different dental services, there are also some medicines for which the country pays a part of the price, for all other medicines you pay the full price), and unless you go to a doctor that has no agreement with the healthcare authority (NFZ) - sometimes you have to wait months or even years for some medical services provided for free, and then you have to pay and go private.