Well, the nearest metro station is Begoña as Jamongris pointed out but...
Those training grounds have been sold to Madrid Municipality and Region (a very complex operation) and four skyscrapers are going to be built there. The works have already started and the main Real Madrid team (i.e. the 'galácticos') no longer train there. If I recall well, since the beginning of this year, they train at the grounds of the Spanish Federation at Las Rozas.
The only remaining building in the old grounds at Paseo de la Castellana (not paseo de castellano
), is the Raimundo Saporta Pavillion, where the Real Madrid basketball team play, and they are waiting for them to end the season in May-June to demolish it.