Post back if you find anywhere that is freehold that you can buy.
Found this :
"GCC nationals may buy residential real estate anywhere in Qatar but they are not permitted to sell it in the first 4 years after purchase. They cannot take out a mortgage loan on the property unless it is in one of the areas designated for foreign investment i.e. the 18 leasehold areas or the 3 freehold areas.
Resident expatriates of all nationalities are entitled to buy freehold or leasehold property in the designated areas. "