We study genus 5 curves C with a fixed point free involution. We give a geometrical (embedded) characterisation of these curves among all genus 5 curves: the points of the Prym variety associated to the involution give embeddings of the curve C in ${{\mathbb P}}_3$ so that C has infinitely many quadrisecant lines. Conversely, any genus 5 curve having such an embedding is endowed with a fixed point free involution and the embedding is given by a point of the Prym variety.