Opera and the Palacio Real are two specific buildings but they also describe the area in which they are located, which is pedestrianised and beautifully preserved. There are gardens and bars which, on Saturdays and Sundays, are full of madrileños promenading and street artists playing music or dancing tango.
The Palacio Real (or Palacio de Oriente) is quite ugly itself, like an oversized wedding cake, but the Jardines de Sabatini are worth investigating. The area takes you to the end of Calle Mayor or to Pintor Rosales on the opposite side.
Metro Opera or Plaza de España