Radiation impacting astronauts in their spacecraft come from a “bath” of high-energy rays (0.1–0.5 mGy per mission day) that reaches deep tissues like the heart and bones and a “stochastic rain” of low-energy particles from the shielding and impacting surface tissues like skin and lenses. However, these two components cannot be reproduced on Earth together. The MarsSimulator facility (Toulouse University, France) emits, thanks to a bag containing thorium salts, a continuous exposure of 120 mSv/y, corresponding to that prevailing in the International Space Station (ISS). By using immunofluorescence, we assessed DNA double-strand breaks (DSB) induced by 1–5 weeks exposure in ISS of human tissues evoked above, identified at risk for space exploration. All the tissues tested elicited DSBs that accumulated proportionally to the dose at a tissue-dependent rate (about 40 DSB/Gy for skin, 3 times more for lens). For the lens, bones, and radiosensitive skin cells tested, perinuclear localization of phosphorylated forms of ataxia telangiectasia mutated protein (pATM) was observed during the 1st to 3rd week of exposure. Since pATM crowns were shown to reflect accelerated aging, these findings suggest that a low dose rate of 120 mSv/y may accelerate the senescence process of the tested tissues. A mathematical model of pATM crown formation and disappearance has been proposed. Further investigations are needed to document these results in order to better evaluate the risks related to space exploration.