When someone is deeply afraid of animals — whether that fear relates to mice, cats, or elephants — they suffer from zoophobia. If you have zoophobia, you probably shouldn't pursue a career as a veterinarian.
This word's roots couldn't be more clear: zoo- relates to animals, from the Greek zoion, and phobia means "irrational fear," also from a Greek root, phobos. Some people with zoophobia are terrified of every non-human animal, although it's more common for zoophobia to relate to a single type of animal, like alligators or lions. In any case, if you suffer from zoophobia, the idea of visiting the zoo likely fills you with dread.