Binding of excess electrons to nanosize water droplets, with a focus on the hitherto largely unexplored properties of doubly-charged clusters, were investigated experimentally using mass spectrometry and theoretically with large-scale first-principles simulations based on spin-density-functional theory, with all the valence electrons (that is, 8e per water molecule) and excess electrons treated quantum mechanically. Singly-charged clusters (H 2O)n-1 were detected for n = 6 - 250, and our calculated vertical detachment energies agree with previously measured values in the entire range 15 ≥ n ≥ 105, giving a consistent interpretation in terms of internal, surface and diffuse states of the excess electron. Doubly-charged clusters were measured in the range of 83 ≥ n ≥ 123, with (H2O)n-2 clusters found for 83 ≥ n < 105, and mass-shifted peaks corresponding to (H2O)n-2(OH -)2 detected for n ≥ 105. The simulations revealed surface and internal dielectron, e-2, localization modes and elucidated the mechanism of the reaction (H2O)n -2 → (H2O)n-2 (OH-)2 + H2 (for n ≥ 105), which was found to occur via concerted approach of a pair of protons belonging to two water molecules located in the first shell of the dielectron internal hydration cavity, culminating in formation of a hydrogen molecule 2H+ + e-2 → H2. Instability of the dielectron internal localization impedes the reaction for smaller (n < 105) doubly-charged clusters.