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").

Edit (bonus): If you're interested in higher dimensions, then let $X$ be the intersection of two quadrics in $\mathbb{P}^{2g+1}$, and let $S$ be the variety of $(g-1)$-planes in $X$.  Then $S$ is a homogeneous space under the Jacobian of a hyperelliptic curve $C$ of genus $g$.  The relationship between $X$, $S$, and $C$ (and the action of $\mathrm{Jac}(C)$ on $S$) is very explicit. Chapter 4 of Miles Reid's PhD thesis (_The complete intersection of two or more quadrics_) has the details.