Hydrogen and helium saturate the 1D pore systems of the high‐silica (Si/Al>30) zeolites Theta‐One (TON), and Mobile‐Twelve (MTW) at high pressure based on x‐ray diffraction, Raman spectroscopy and Monte Carlo simulations. In TON, a strong 22 % volume increase occurs above 5 GPa with a transition from the collapsed P2$_1$ to a symmetrical, swelled Cmc2$_1$ form linked to an increase in H$_2$ content from 12 H$_2$/unit cell in the pores to 35 H$_2$/unit cell in the pores and in the framework of the material . No transition and continuous collapse of TON is observed in helium indicating that the mechanism of H$_2$ insertion is distinct from other fluids. The insertion of hydrogen in the larger pores of MTW results in a strong 11 % volume increase at 4.3 GPa with partial symmetrization followed by a second volume increase of 4.5 % at 7.5 GPa, corresponding to increases in hydrogen content from 43 to 67 and then to 93 H$_2$/unit cell. Flexible 1D siliceous zeolites have a very high H$_2$ capacity (1.5 and 1.7 H$_2$/SiO$_2$ unit for TON and MTW, respectively) due to H$_2$ insertion in the pores and the framework, in contrast to other atoms and molecules, thereby providing a mechanism for strong swelling.