Which field of anthropology studies the hunting and gathering techniques of early man?

The correct answer is Archeology. The other subcategories of Anthropology as the study about humans are Cultural Anthropology, Biological Anthropology and Linguistic Anthropology.

The study of human activity which includes biofacts, artifacts and cultural activities like early hunting, is a branch of humanities and also regarded as a social science, is Archaeology and especially Prehistoric Archaeology, which stretches back to ancient eras of human existence through the exploration of remains.