  In this paper we compute the dimension of a class of dynamically defined
non-conformal sets. Let $X\subseteq\mathbb{T}^2$ denote a Bedford-McMullen set
and $T:X\to X$ the natural expanding toral endomorphism which leaves $X$
invariant. For an open set $U\subset X$ we let X_U={x\in X : T^k(x)\not\in U
\text{for all}k}. We investigate the box and Hausdorff dimensions of $X_U$ for
both a fixed Markov hole and also when $U$ is a shrinking metric ball. We show
that the box dimension is controlled by the escape rate of the measure of
maximal entropy through $U$, while the Hausdorff dimension depends on the
escape rate of the measure of maximal dimension.
