The moduli space of \'etale double covers of genus 5 curves is   unirational
  We show that the coarse moduli space $\cR_5$ of \'etale double covers of curves of genus~5 over the complex numbers is unirational. We give two slightly different arguments, one purely geometric and the other more computational. 
E. Izadi, M. Lo Giudice and G.K. Sankaran