It's impossible for anyone to definitively say they don't exist. That would require proving a negative, which is logically impossible. However, considering there is no evidence whatsoever that they exist, and such a being would be so far outside the realm of everything we know about biology and species diversity, there is no reason to believe they are real or that there is even a slight chance that they are. But it's a really big universe, so if you extend the possibility of the existence of fairies to any other planet in the universe that could support life then your odds would go up, at least for their existence as a biological life form with a small humanoid body and wings for flight. The whole notion of them having magic, however, would defy the laws of physics.