Totally agree with mecabq. Souq Waqif is the place to go during a short visit. It IS the place where Qataris go in the evening (as well as expats). You won't see a Qatari on the street (in general). And I don't think a visitor that comes for a few hours wants to go in places like QBC.

If you count workers form Asia (India, Bangladesh, Sri Lanka etc. who are the majority of country's population) in locals then a good place to see their life is around Mercury hotel which is right next to Souq Waqif.