They are neither fruit nor vegetable. They are considered berries. As are watermelons, pineapples, oranges and lemons. And as long as we're on the subject, Botanically speaking, the banana is a herb, the jalapeno is a fruit and apples and peaches are a member of the rose family. So now you know.
Technicaly, by classification, they are fruits but by common social agreement theyre considered veggies. Same with anything in that family including cucumber, egg plants and squash. Interesting FYI - tomatoes are members of the deadly nightshade family and are supposedly poisonous when green. Not that ive ever heard of anyone dying from fried green tomatos or tomato relish or chow-chow or pickled green tomatos. Dont eat the leaves tho.