Roosevelt was a peripatetic traveler, covering large distances in service of the many projects she pursued both during and after the FDR presidency. It was Eleanor’s determination to drive her own car that led to her pistol ownership. The Secret Service begged her to take an agent, a police escort, or at least a chauffeur; she refused. The pistol was a compromise: a small bit of protection to put their minds at ease...Found in Slate's Vault. I doubt this is who Bing Crosby had in mind for his song.
Although Eleanor told the fascinated press, when she first got the weapon, that she was a “fairly good shot,” a New York Times reporter at the 1972 dedication of the Eleanor Roosevelt Wings of the FDR Library interviewed several of Eleanor’s friends who said that she carried the permit, but not the pistol.