There's no One True Definition of what it means to be fractal, but a lot of working mathematicians use the criterion that a space's Hausdorf dimension is not equal to its topological dimension. Oftentimes (but not always), the Hausdorf dimension will be fractional, which is where the word comes from. Self-similarity is an easy way to satisfy that requirement in a way that's easy to explain, but it's far from the only way.