Time is like a clock , it's cyclical. My inspiration for this is like everything else it's art and worlds drawn by the furry community at large and explains the whats and whys living eternally is even feasible for my characters.
For example, let's take Caroo, his OC shows up in many different time periods and it wouldn't normally be possible without the help of time travel, but the idea is the same, Caroo is immortal and there'll be a point where technology gets annihilated and civilizations are spread far apart, so Caroo will experience the prehistoric times. The same thing was explored in the latest zelda game where it's implied the kingdom of hyrule was destroyed and regressed into a more traditional civilization with the zonai.
All my cartoon characters may have lived for decades, centuries, millenia even without realizing it, but lived through what normies would call a number of generations, shifts in technologies and cultures. They may even have at one point been jungle dwellers who been hypnotized if not eaten by a snake.
2 things of note: Their own version of the big bang, whether real or not has never been conceptualized or imagined and there is no galaxy cooling or universe expansion that's going to cause their reality to stop existing or say an end of times, but it's still possible maybe? Even the smartest cartoon savants have deduced that everythig in the universse is perfectly stable and unlikely to change unlike our reality.
In short, in the cartoon world there is seemingly no start and no end to time itself.