In this paper, we study some cube packing problems. In particular, we are interested in compact subsets of ℝ[sup n], n ≥ 2, which contain boundaries of cubes with all side lengths in (0,1). We show here that such sets must have lower box dimension at least n - 0.5, and we will also provide sharp examples. We also show here that such sets must be large in general in a precise sense which is also introduced in this paper.