  Let $R>1$ and let $B$ be the Euclidean $4$-ball of radius $R$ with a closed
subset ${E}$ removed. Suppose that $B$ embeds symplectically into the unit
cylinder $\mathbb{D}^2 \times \mathbb{R}^2$. By Gromov's non-squeezing theorem,
${E}$ must be non-empty. We prove that the Minkowski dimension of ${E}$ is at
least $2$, and we exhibit an explicit example showing that this result is
optimal at least for $R \leq \sqrt{2}$. In an appendix by Jo\'e Brendel, it is
shown that the lower bound is optimal for $R < \sqrt{3}$. We also discuss the
minimum volume of ${E}$ in the case that the symplectic embedding extends, with
bounded Lipschitz constant, to the entire ball.
