We construct new fractal acoustic metamaterials by coiling up space, which can allow subwavelength-scale and broadband sound insulation to be achieved. Using the finite element method and the S-parameter retrieval method, the band structures, the effective parameters, and the transmission losses of these acoustic metamaterials with different fractal orders are researched individually. The results illustrate that it is easy to form low-frequency bandgaps using these materials and thus achieve subwavelength-scale sound control. As the number of fractal orders increase, more bandgaps appear. In particular, in the X direction of the acoustic metamaterial lattice, more of these wide bandgaps appear in different frequency ranges, thus providing broadband sound insulation and showing promise for use in engineering applications.