Self-similar structures occur naturally and have been employed to engineer exotic physical properties. We show that acoustic modes of a fractal-like system of tensioned strings can display increased mechanical quality factors due to the enhancement of dissipation dilution. We describe a realistic resonator design in which the quality factor of the fundamental mode is enhanced by as much as 2 orders of magnitude compared to a simple string with the same size and tension. Our findings can open new avenues in force sensing, cavity quantum optomechanics, and experiments with suspended test masses.