We construct new quasi-three-dimensional fractal acoustic metamaterials based on adoption of the Menger structure, which offers extraordinary parameters such as double-negative properties and a near-zero density. The resulting metamaterials can thus achieve negative refraction, acoustic focusing and sound tunneling. Using the finite element method and the S-parameter retrieval method, the band structures and the effective parameters of these acoustic metamaterials are researched, respectively. The negative refraction property is numerically simulated using a Gaussian beam passing through a double negative prism. A plate lens with a refractive index of n?=??1 is constructed to achieve acoustic focusing and the sound tunnelling ability is verified using the near-zero-density metamaterial. The results show that the Menger fractal structures have excellent acoustic properties and are promising for acoustic applications.