The principle of hierarchical design is a prominent theme in many natural systems where mechanical efficiency is of importance. Here we establish the properties of a particular hierarchical structure, showing that high mechanical efficiency is found in certain loading regimes. We show that in the limit of gentle loading, the optimal hierarchical order increases without bound. We show that the scaling of material required for stability against loading to be withstood can be altered in a systematic, beneficial manner through manipulation of the number of structural length scales optimized upon. We establish the relationship between the Hausdorff dimension of the optimal structure and loading for which the structure is optimized. Practicalities of fabrication are discussed and examples of hierarchical frames of the same geometry constructed from solid beams are shown.