I will introduce the new notion of R*-category (arXiv:2312.02883) and explain the ways in which it is a good answer to the question in the title. The main difference between R*-categories and abelian categories is extra structure that generalises adjoints of morphisms: R*-categories come with a functorial and involutive choice of morphisms f*:Y → X for each morphism f: X → Y. Many elementary aspects of the theory of Hilbert spaces generalise to arbitrary R*-categories, including the relationship between positive operators and contractions. Familiarity with only the most basic concepts from category theory (categories, products, equalisers, monomorphisms) will be sufficient to understand most of the talk.