We are a family of 4 looking for a central madrid close to activity but not right in the middle of the tourist noise - looking for a neighborhood with the charm of the village in new york for example.
We did a VRBO on a third floor apartment overlooking the Plaza Mayor. Noise wasn't bad and location was very good.
Look around the area bounded in the north by Plaza Mayor and Plaza Sta Ana, and to the south by metros La Latina and Anton Martin.
The land of Oz continues to resonate with great recommendations as we always stay in an apt in the latina district. It offers a short walk to the main tourist area and be sure to find the market and the meat shops are terrific.
La Latina is my favorite. Don't overlook Malasana and Chueca, though. A little farther out, but still very close to the center. Lots of tapas bars and groceries in all 3 neighborhoods.