A zoo used to be little more than animals in cages put on display for public viewing. The idea is a modernization of the menageries that used to be kept by nobility, and the first public zoos were just these menageries opened to the public:

Until the early 19th century, the function of the zoo was often to symbolize royal power, like King Louis XIV's menagerie at Versailles. Major cities in Europe set up zoos in the 19th century, usually using London and Paris as models. The transition was made from princely menageries designed to entertain high society with strange novelties into public zoological gardens. (ref)

Depends on the animal. I know a decent zoo that has just simple waist-high walls around the porcupines and raccoon dogs for example.

They have space, but they can get pretty close. I guess a complete idiot with a cane could probably harass them, until someone stops them like "what the hell, asshole". Or you know, until Everett True comes and causes permanent spine damage to them.

Obviously animals like bears, apes and big felines can't get that close to people.