Just a piece of advice. If you do decide to go over to the second-hand car market that gets held during weekends, be very careful. You will find several people who just buy and sell cars pocketing a small profit. This is their business. Many are from car shops that resell second-hand cars. Try looking for a "genuine" seller, someone who is possibly leaving the country. I believe Qatar Living to be good site for such sellers.
Keep a simple rule in mind: Buying a second-hand car is almost always a "blind buy." That is why it is always advisable to buy from a person who is leaving the country, especially from a person who has been a single owner of the car.
There are second-hand car markets held on weekends at some locations in Doha. One place is in the Industrial city area but I am not sure of the location. If you ask your friends and colleagues, they may be knowing. Most Taxi drivers should also be able to give you the location and the timing. I guess the market works only in the afternoons usually after Juma'a prayers.