I have a friend from high school who is a veterinarian. She helps out at local farms as an extracurricular activity. In addition, she provides free care for older horses and supports local animal events like free shots, adoptions, and wellness education. Growing up, she offered to babysit or house sit for all types of animals so that she could learn about how to care for them. She even offered to help local vet's wash their animals and clean out cages and pins. She went to college in Colorado as they have one of the best veterinary schools in the country. Hope this helps.
That's a really broad question. To be considered extracurricular for a veterinarian, in my opinion it should be non-vet related, otherwise, it's just an extension of their vet practice. For example providing free vet services is still "being a vet" so I wouldn't consider it extracurricular.
One vet I know likes to quilt; another likes to run marathons; another shows dogs; another likes to golf, another likes to surf; one is a pilot, etc. As you can see, they are all over the place with their extracurricular activities. There are really no boundaries, it's all just a matter of personal interest.
Now if you're talking about a good extracurricular activity for a veterinary student, then you should look for anything animal related like: Pet sitting; volunteering/interning at a zoo or vet clinic; volunteering at the animal shelter, etc.