Someone or something invulnerable cannot be harmed or damaged, even if you attack them or speak against them. In stories, a castle or a hero may be invulnerable. In ordinary English we mostly use it for feelings and reputation: invulnerable to criticism, invulnerable to doubt. It does not mean the person will live for ever — that is immortal. It means this kind of harm does not land. Though actually, we often say it about a feeling of safety that may not be true: “He felt invulnerable” usually prepares you for a fall. It is a fairly formal adjective; everyday talk is more often “nothing can hurt them.”