`Oumuamua (I1 2017) was the first macroscopic ($l\sim100\,{\rm m}$) body observed to traverse the inner solar system on an unbound hyperbolic orbit. Its light curve displayed strong periodic variation, and it showed no hint of a coma or emission from molecular outgassing. Astrometric measurements indicate that 'Oumuamua experienced non-gravitational acceleration on its outbound trajectory, but energy balance arguments indicate this acceleration is inconsistent with a water ice sublimation-driven jet of the type exhibited by solar system comets. We show that all of `Oumaumua's observed properties can be explained if it contained a significant fraction of molecular hydrogen (H$_{2}$) ice. H$_{2}$ sublimation at a rate proportional to the incident solar flux generates a surface-covering jet that reproduces the observed acceleration. Mass wasting from sublimation leads to monotonic increase in the body axis ratio, explaining `Oumuamua's shape. Back-tracing `Oumuamua's trajectory through the Solar System permits calculation of its mass and aspect ratio prior to encountering the Sun. We show that H$_{2}$-rich bodies plausibly form in the coldest dense cores of Giant Molecular Clouds, where number densities are of order $n\sim10^5$, and temperatures approach the $T=3\,{\rm K}$ background. Post-formation exposure to galactic cosmic rays implies a $\tau \sim 100$ Myr age, explaining the kinematics of `Oumuamua's inbound trajectory.