They aren't speaking literally. From my point of view, a human who can live, say, 10,000 years is as good as 'immortal'. Sure its literally wrong, but I heard webster now defines 'literally' as 'figuratively' so I'm ready to give up on pedantry.
I read a lot of Niven's short stories as a child, but I don't remember that title. Did he illustrate some of the problems we'd have with long life spans?
Not really. It was more of an exploration into different perspectives on search for life-extending technologies by aliens with different life spans. I suggest reading it.
Even if we had a "permanent" energy source, we have only a finite number of atoms. So one day we would actually also be guaranteed to cycle our thoughts.