  We study the geometry of domains in complete metric measure spaces equipped
with a doubling measure supporting a $1$-Poincar\'e inequality. We propose a
notion of \emph{domain with boundary of positive mean curvature} and prove
that, for such domains, there is always a solution to the Dirichlet problem for
least gradients with continuous boundary data. Here \emph{least gradient} is
defined as minimizing total variation (in the sense of BV functions) and
boundary conditions are satisfied in the sense that the \emph{boundary trace}
of the solution exists and agrees with the given boundary data. This extends
the result of Sternberg, Williams and Ziemer to the non-smooth setting. Via
counterexamples we also show that uniqueness of solutions and existence of
\emph{continuous} solutions can fail, even in the weighted Euclidean setting
with Lipschitz weights.
