Jacobians of genus-2 curves - and abelian surfaces in general, I suppose - can be realized as the variety of lines on the intersection of two quadrics in $\mathbb{P}^5$ (once you've chosen a line to act as the neutral element).  This is analogous to seeing an elliptic curve as the variety of 0-dimensional spaces (i.e. points) on the intersection of two quadrics in $\mathbb{P}^3$ (which is sometimes called the "Jacobi" model of an elliptic curve).  The group law has a really nice geometric expression.

This is covered at length in Chapter 17 ("A neoclassical approach") of Cassels and Flynn's _Prolegomena to a middlebrow arithmetic of curves of genus 2_, and in even more length in Chapter 6 of _Principles of algebraic geometry_ by Griffiths and Harris (specifically Section 6.3, "Lines on the quadric line complex").