expatriate, I think you'll find a number of threads about bicycle ownership in Madrid on this message board - just conduct a search. But it's really not recommended because it can be quite dangerous competiting with cars, the saturation of sidewalks with pedestrians, high rate of bicycle theft when they're parked/chained outside, etcetera.

Maybe people on Spain's www.SegundaMano.es sells used bicycles. Give them a try.

Best of luck! Saludos, MadridMan