Mr. Bemis expects to live "years and years and years" post-attack. In 1959, the effects of nuclear fallout were still under initial study and not widely understood. We see Bemis emerging from the bank vault more or less immediately after a thermonuclear blast (bomb-shelter protocol says to wait at least two weeks for the worst of the radioactivity to decay), eating packaged food that was probably irradiated, etc. So even with his glasses, Bemis would have not lived to enjoy his books for very long. See more
The best-laid plans of mice and men - and Henry Bemis, the small man in the glasses who wanted nothing but time. Henry Bemis, now just a part of a smashed landscape, just a piece of the rubble, just a fragment of what man has deeded to himself. Mr. Henry Bemis - in the Twilight Zone.