This article is devoted to some fundamental aspects of “super” storage in graphite nanofibers (GNF) of “reversible” (~20–30 wt.%) and “irreversible” hydrogen (~7–10 wt.%). Extraordinary results for hydrogen “super” storage were previously published by the group of Rodriguez and Baker at the turn of the century, which been unable to be reproduced or explained in terms of physics by other researchers. For the first time, using an efficient method of processing and analysis of hydrogen thermal desorption spectra, the characteristics of the main desorption peak of “irreversible” hydrogen in GNF were determined: the temperature of the highest desorption rate (Tmax = 914–923 K), the activation energy of the desorption process (Q ≈ 40 kJ mol−1), the pre-exponential rate constant factor (K0 ≈ 2 × 10−1 s−1), and the amount of hydrogen released (~8 wt.%). The physics of hydrogen “super” sorption includes hydrogen diffusion, accompanied by the “reversible” capture of the diffusant by certain sorption “centers”; the hydrogen spillover effect, which provides local atomization of gaseous H2 during GNF hydrogenation; and the Kurdjumov phenomenon on thermoelastic phase equilibrium. It is shown that the above-mentioned extraordinary data on the hydrogen “super” storage in GNFs are neither a mistake nor a mystification, as most researchers believe.