For C a smooth curve and ξ a line bundle on C, the moduli space UC(2, ξ) ofsemistable vector bundles of rank two and determinant ξ is a Fano variety. We show that UC(2, ξ) is K-stable for a general curve C ∈ Mg. As a consequence, there are irreducible components of the moduli space of K-stable Fano varieties that are birational to Mg. In particular these components are of general type for g ≥ 22.