Would it be possible? It strikes me as impossible but I am still curious to know what you think.
A story is a narrative - an account of connected events. Somebody is giving that account - there's no avoiding that. There you've got your narrator. Even a newspaper, which seeks to make the journalist impartial and transparent, there's still the person reporting on what happened, recounting it, narrating it. Even dialogue: the moment there's "Alice said", "Bob replied", it's the narrator telling you who said what.
Now, what happens if there's only the dialogue, no narration in between? Then you've got a play. To be fair, I've read significantly more plays than I've seen, but that doesn't turn a play into a novel.
Not really, even if the narrator is completely invisible to the reader, and not actually a person, as in a third person omniscient narrative, there is still a viewpoint from which the story is told. That viewpoint is still the narrator regardless of its material existence or its (non)existence within the narrative it is relating. Furthermore people will tend to anthropomorphise that viewpoint anyway, for example I just had to rewrite that whole second sentence with "it" instead of "they".