  We study geometric properties of compact stable minimal surfaces with
boundary in homogeneous 3-manifolds $X$ that can be expressed as a semidirect
product of $\mathbb{R}^2$ with $\mathbb{R}$ endowed with a left invariant
metric. For any such compact minimal surface $M$, we provide a priori radius
estimate which depends only on the maximum distance of points of the boundary
$\partial M$ to a vertical geodesic of $X$. We also give a generalization of
the classical Rado's Theorem in $\mathbb{R}^3$ to the context of compact
minimal surfaces with graphical boundary over a convex horizontal domain in
$X$, and we study the geometry, existence and uniqueness of this type of
Plateau problem.
