Your Government Health Card entitles you to heavily subsidised treatment at state hospitals like the Hamad Hospital. It's not free, but the fees are very low. You can check prices at www.hmc.org.qa. Yes, there are plenty of private hospitals and clinics, but you will pay a minimum of 200 QAR per visit. You need to find out if your company will reimburse this. The health card is definitely worth having, it entitles you to cheap healthcare, and the hospitals are very well equipped. However, you will be subject to availability of doctors and waiting time, same as in any state-run hospital. Waiting is almost non-existant in the private hospitals. As for insurance, health insurance is only just catching on in Qatar but I believe it is provided by QIC - Qatar Insurance Company.