Does South Korea have a national health insurance?
Yes. Its official name in English is National Health Insurance Service, or NHIS for short. In Korean, it's called guhn-gahng boh-huhm meaning 'health insurance.' It is required by law for all Korean citizens to be insured to this service.
Unfortunately, this insurance is not free. If you are a Korean citizen with a job, you pay half and your employer pays the other half to make it 4.48% of your wage. If you are not employed, they make a calculation of your assets and other factors to determine how much you are supposed to pay.
Some foreigners may be able to be insured as well. According to my knowledge, you pay around 50 dollars every month. But there is a catch. Let's say you came to Korea two years ago. If you want to be insured, you must pay for the entire duration of your visit, starting on the entry date of your last arrival to South Korea. So if you arrived two years ago, the amount will be $1,200 ($50 x 24 months). Once you pay $1,200 up front, you will be insured within a few days.