I remember the academy well. My grandma lived a few blocks away on 10th near Balboa, and every week my mom and I would visit her and walk to Children's Playground in the park. On the way back, we always stopped at the stables so I could see the horses.
I must have been very young (4 or 5?) but still retain vivid distinct memories of the place: cobblestone or brick paving; wood shavings and hay; a long corridor with stalls opening off it and horse heads bobbing in the shadows; and dark wood paneling everywhere.
And of course, the smell. Anyone who's ever worked around a stable knows the odor: horse manure, urine, leather tack, and alfalfa and dry feed intermingled. Surprisingly, it wasn't unpleasant -- even to a toddler.
The indoor rink was magical; we'd sit in the seats and watch the "horsies" cantering in circles as light filtered down through windows probably unwashed for decades. Even then I knew I was visiting a place out of the City's past.