Solids with a hierarchically structured, disordered pore space, such as macroporous-mesoporous silica monoliths, are used as fixed beds in separation and catalysis. Targeted optimization of their functional properties requires a knowledge of the relation among their synthesis, morphology, and mass transport properties. However, an accurate and comprehensive morphological description has not been available for macroporous-mesoporous silica monoliths. Here we offer a solution to this problem based on the physical reconstruction of the hierarchically structured pore space by nanoscale tomography. Relying exclusively on image analysis, we deliver a concise, accurate, and model-free description of the void volume distribution and pore coordination inside the silica monolith. Structural features are connected to key transport properties (effective diffusion, hydrodynamic dispersion) of macropore and mesopore space. The presented approach is applicable to other fixed-bed formats of disordered macroporous-mesoporous solids, such as packings of mesoporous particles and organic-polymer monoliths.