There are some anime characters who are just straight-up impossible to eliminate. You can chop off their heads, jab them in the heart, or pump them full of toxins, and they just won't perish. Often, this is because they're immortal. Maybe they drank an elixir like the immortals did in Baccano! or were merely born indestructible, such as Kei Nagai of Ajin, but no matter what the reason is, loss of life is off the table for these characters.
Not all anime characters you can't kill are immortal - while the vast majority are, there are a few like Saitama of One-Punch Man, who are so powerful that the idea of someone taking their life seems impossible.
In some cases, the inability to perish is a blessing, but more often than not, it's a curse. Alucard of Hellsing is happy about his infinite lifespan - all the more time to exercise his bloodlust - but Joseph of The Ancient Magus' Bride is so desperate to expire that he's turned to human experimentation to make it happen.
Whatever the situation, these unkillable characters are often part of fascinating stories that question the true value of eternal life and invulnerability.
Ban is immortal, which means he can't perish or age; he also can miraculously heal himself. He gained immortality by seeking out the Fountain of Youth. The guardian of the fountain, Elaine, didn't want to allow him to drink from it at first because she believed he had bad intentions, but she became moved when he explained his terrible life and how he thinks that if he lives long enough, it will improve. When demons swoop in, she ends up giving him the water via a kiss to save his life.
His immortality serves him well throughout the series - it saves him from multiple execution attempts, but sometimes he doesn't feel truly alive, so he seeks out physical agony.
In the world of Hellsing, immortality does not exist per se, but Alucard is about the closest thing to it. He's a vampire, and when he sucks blood, he doesn't only get a good dose of iron - he also sucks out his victims' souls, which adds substantially to his own lifespan.
He's also so absurdly powerful that only the strongest opponent could even put a dent in him.
Saitama isn't technically immortal, but nothing he's encountered so far - including a giant meteor hurtling right toward him - has come anywhere close to offing him, so he may as well be indestructible. Saitama can easily defeat anyone with a single punch, so he's unlikely to get taken out in the midst of battle.
The only thing that could possibly slay him? Stress - after all, the guy worked out so hard he lost all his hair.
As a Death God, Ryuk can only pass in one of two ways - he stops writing names in his Death Note and adding human lifespans onto his own, or he does something with the intention of saving a human life. Ryuk has shown no inclination toward valuing human life for anything other than his own amusement, and he's not lazy enough to perish from a simple failure to add to his lifespan.
As long as Ryuk doesn't screw it up himself, he'll live for eternity.