  We give a complete characterization of those disk bundles over surfaces which
embed as rationally convex strictly pseudoconvex domains in $\mathbb{C}^2$. We
recall some classical obstructions and prove some deeper ones related to
symplectic and contact topology. We explain the close connection to Lagrangian
surfaces with isolated singularities and develop techniques for constructing
such surfaces. Our proof also gives a complete characterization of Lagrangian
surfaces with open Whitney umbrellas, answering a question first posed by
Givental in 1986.
