We show by both experiment and theory that a specific class of planar conducting fractals possesses a series of self-similar resonances, leading to multiple gaps and pass bands for electromagnetic waves over an ultrawide frequency range. A double stack of these fractal patterns exhibits polarization-independent absolute gaps over a wide range of incidence angles. These characteristics are retained even when the fractal patterns are significantly subwavelength in all dimensions. In addition, the transmittance can be modulated by an external current source.