Plain and simple, it means not having to die. The fear and the enigma of death is one of the most ancient and persisting human obsessions, that has been the subject of countless mythological stories and written pieces of literature.
In classic Greek and Roman literature, gods were immortal by definition. But there is also an interesting story about Cumaean Sybil, a priestess who was pursued by Apollo. When he promised to grant her one wish if she would yield to him, she asked to live eternally. But then she changed her mind and refused Apollo, who left her to decay alive, as she didn't ask for eternal youth.
In contemporary literature, there are numerous examples of fascination with the so-called "undead" - vampires, aliens, and other uncanny creatures.
