The initial isotopic composition of water in the Solar System is of paramount importance to understanding the origin of water on planetary bodies but remains unknown, despite numerous studies. Here we use the isotopic composition of hydrogen in calcium–aluminium-rich inclusions (CAIs) from primitive meteorites, the oldest Solar System rocks, to establish the hydrogen isotopic composition of water at the onset of Solar System formation. We report the hydrogen isotopic composition of nominally anhydrous minerals from CAI fragments trapped in a once-melted host CAI. Primary minerals have extremely low D/H ratios, with δD values down to −850‰, recording the trapping of nebular hydrogen. Minerals rich in oxidised iron formed before the capture of the fragments record the existence of a nebular gas reservoir with an oxygen fugacity substantially above the solar value and a D/H ratio within 20% of that of the Earth’s oceans. Hydrogen isotopes also correlate with oxygen and nitrogen isotopes, indicating that planetary reservoirs of volatile elements formed within the first 2 × 10$^5$ years of the Solar System, during the main CAI formation epoch. We propose that the isotopic composition of inner Solar System water was established during the collapse of the protosolar cloud core owing to a massive admixture of interstellar water.