  We analyze GIT stability of nets of quadrics in $\mathbb{P}^4$ up to
projective equivalence. Since a general net of quadrics defines a canonically
embedded smooth curve of genus five, the resulting GIT quotient gives a
birational model of the moduli space of genus 5 curves. We study the geometry
of the associated contraction and prove that the constructed GIT quotient is
the final step of the log minimal model program for the moduli space of genus 5
curves.
