A place where bears are kept.
In Elizabethan times, where bears were kept for sport, that is, bears were pitted against each other, or in default of that, dogs, humans, or any other kind. In the Winter's Tale, a character is pursued by a bear, this was supposed to have been supplied by the Garden immediately adjacent to the Theater.
In centuries hence, a Bear Garden is supposed to be a place of noise and tumult, and therefore to be shunned, hence Mary Poppins's line "Is this a Bear Garden?", a way for P. L. Travers to hint that Mary wasn't quite as young as she looked...by like about two millennia?