We provide direct experimental evidence for a non-random distribution of atomic constituents in Zr-based multi-component bulk metallic glasses using positron annihilation spectroscopy. The Ti content around the open-volume regions is significantly enhanced at the expense of Cu and Ni, indicating that Cu and Ni occupy most of the volume bounded by their neighboring atoms while Ti and Zr are less closely packed and more likely to be associated with open-volume regions. Temperature-dependent measurements indicate the presence of at least two different characteristic sizes for the open volume regions. Measurements on hydrogen- charged samples show that the larger open-volume regions can be filled by hydrogen up to a critical density. Beyond this critical density, local atomic-scale open-volume damage is created in the sample to accommodate additional hydrogen. The onset of this local damage in positron annihilation data coincides with the onset of volume expansion in X-ray diffraction data.