  In the setting of a metric space equipped with a doubling measure supporting
a $(1,1)$-Poincar\'e inequality, we study the problem of minimizing the
BV-energy in a bounded domain $\Omega$ of functions bounded between two
obstacle functions inside $\Omega$, and whose trace lies between two prescribed
functions on the boundary of $\Omega.$ If the class of candidate functions is
nonempty, we show that solutions exist for continuous obstacles and continuous
boundary data when $\Omega$ is a uniform domain whose boundary is of positive
mean curvature in the sense of Lahti, Mal\'y, Shanmugalingam, and Speight
(2019). While such solutions are not unique in general, we show the existence
of unique minimal solutions. Our existence results generalize those of Ziemer
and Zumbrun (1999), who studied this problem in the Euclidean setting with a
single obstacle and single boundary condition.
