According to this new travel piece in the National, here are two possible places:

Heritage House near Rumailah Park, opposite the Corniche – something of an undiscovered treasure. No more than 10 Qatari women work every day perfecting traditional Khaleeji handicrafts from 7am until 1.30pm in a majlis-style open space. Entrance and a guided tour is free.

Souq Waqif – Squashed between the international restaurants and chains that fill the refurbished bazaar, local chefs cook for a largely local clientele. The restaurants have Arabic signs, food is served in silver foil containers and Qatari citizens fill the wooden benches.

Anyone else have thoughts?