There are two types of insurance companies in Korea. You can apply to whichever fits your situation best.
Korean National Health Insurance:, Tel.+82-1577-1000 You pay around 36,000 monthly and can receive hospital aids just like the assistance Korean citizens receive. You can personally apply for this insurance at the Kukmin Health Insurance Industrial Complex nearest you.
Commercial insurance company: These are the general insurance companies that sell accident insurances at various prices. However, you are advised to carefully study their terms and conditions. 
Health insurance1 Health insurance2