Eternity Lost

Story, first published: Astounding Science Fiction, July 1949

The theme of the consequences of becoming immortal was always in the back of Simak's mind. ... In "Eternity Lost" ..., Simak was concerned about having enough room for all the people when immortality becomes a reality. When pioneers are sent to the stars in this story, they are granted the gift of immortality, while others remaining on Earth (If worthy of the privilege) merely have their lives extended.

Ewald, Robert J.: When the Fires Burn High and the Wind is from the North, p.89

This future earth has rejuvenation treatments called "life continuation" - a single treatment gives you youthfulness & extends your life by a century. But it cannot be widely made available, because earth doesn't have resources to support so many long lived people.
So only a few important people receive the treatment, & repeatedly - "those persons who are in the unique position of being able to materially benefit the human race." No prizes for guessing who the privileged ones are.
Officially, there are strict rules for determining who's important enough; but what are the rules for if they cannot be bent?
One of the men who has received this treatment 5 times is Senator Homer Leonard, an elected politician representing some US constituency in World House of Representatives in Geneva. He was the key legislature when current rules for determining beneficiaries of rejuvenation treatments were framed.
It's time for his next treatment. It's his (political) party that has been sponsoring him for treatment so far, declaring him important enough. Now, finally, they're not going to sponsor him. He's guessing the reasons have to do with his decreasing ability to get votes.
Because of his inside knowledge, he suspects that even immortality treatments are available. But they're not given to anyone known - limited resources on earth, etc. An (apparently) publicly funded organization called Extrasolar Research is looking for earth-like worlds among the stars - may be someday everyone will have immortality. And he has a suspicion such an alternate earth is about to be found - probably before his next rejuvenation is due. If only he could get it this one last time!
He will make pathetic (& unsuccessful) efforts at getting the treatment; in the process, we get a view of how much ordinary people resent the recipients of this treatment.
Eventually, angry with his party & to score some political points, he makes a press announcement that he go for any more rejuvenation treatment - something that no beneficiary has ever done before. But there is one little thing he'd not found in time - he wasn't really out of favor. Not only is immortality treatment available, an alternate earth has been found & he was chosen among the early colonists - only he's lost his chance, through his own stupidity.

Tinkoo Valia: Variety SF