Acorns are both botanically and culinarily nuts. Botanically, they have single seed in a shell which doesn’t break open along a set line, making them nuts scientifically.
In a practical sense, I think most everyone would consider acorns to be nuts. It is quite a stereotypical nut for squirrels to eat in cartoons. Many people may not really even think about acorns in a human-food sense, but acorns have been eaten for food. This was most common among the American Indians, but peoples throughout Europe and Asia have also eaten them. Regardless of how appetizing you find acorns, they are pretty incontrovertibly nuts.