Hydrogen-bond networks play vital roles in biological functions ranging from protein folding to enzyme catalysis. Here we combine electronic structure calculations and ab initio path integral molecular dynamics simulations, which incorporate both nuclear and electronic quantum effects, to show why the network of short hydrogen bonds in the active site of ketosteroid isomerase is remarkably robust to mutations along the network and how this gives rise to large local electric fields. We demonstrate that these properties arise from the network's ability to respond to a perturbation by shifting proton positions and redistributing electronic charge density. This flexibility leads to small changes in properties such as the partial ionization of residues and pICa isotope effects upon mutation of the residues, consistent with recent experiments. This proton flexibility is further enhanced when an extended hydrogen-bond network forms in the presence of an intermediate analogue, which allows us to explain the chemical origins of the large electric fields in the enzyme's active site observed in recent experiments.