In 2017, 1I/`Oumuamua was identified as the first known interstellar object in the Solar System1. Although typical cometary activity tracers were not detected2-6, `Oumuamua showed a notable non-gravitational acceleration7. So far, there has been no explanation that can reconcile these constraints8. Owing to energetic considerations, outgassing of hyper-volatile molecules is favoured over heavier volatiles such as H2O and CO2 (ref. 9). However, there are theoretical and/or observational inconsistencies10 with existing models invoking the sublimation of pure H2 (ref. 9), N2 (ref. 11) and CO (ref. 12). Non-outgassing explanations require fine-tuned formation mechanisms and/or unrealistic progenitor production rates7,13-15. Here we report that the acceleration of `Oumuamua is due to the release of entrapped molecular hydrogen that formed through energetic processing of an H2O-rich icy body. In this model, `Oumuamua began as an icy planetesimal that was irradiated at low temperatures by cosmic rays during its interstellar journey, and experienced warming during its passage through the Solar System. This explanation is supported by a large body of experimental work showing that H2 is efficiently and generically produced from H2O ice processing, and that the entrapped H2 is released over a broad range of temperatures during annealing of the amorphous water matrix16-22. We show that this mechanism can explain many of `Oumuamua's peculiar properties without fine-tuning. This provides further support3 that `Oumuamua originated as a planetesimal relic broadly similar to Solar System comets.