I suppose it's moot now for the OP, but for others who read this in the future, as BiziBB said, the five-hour rule doesn't make sense. I doubt that they would ask you (especially if you have a passport from a Western or wealthy Arab country) what you are doing in Doha, how long you plan to stay, etc. They have never asked me this in my many times entering; they might ask you where you are staying, in which case just tell them the name of the hotel that you are going to for your drinks and dinner. (The visa will cost QAR 100, or about $27, per person.)

The Marriott and Sharq Village (a Ritz-Carlton property) are close to the airport (probably 10-20 minutes by taxi). Five hours would be cutting it close, but is certainly enough time to take a taxi and have a leisurely dinner and drinks. Alternatively, you could go to the Souq Waqif, the major bazaar, which has lots of shops and restaurants (although no alcohol) and is a good place to kill a few hours and get a bit of local flavor.

By the way, calinhove, thanks for the tip alcohol in the lounge -- it never would have occurred to me to ask!