Zero-refractive-index materials may lead to promising applications in various fields. Here, we design and fabricate a near Zero-Refractive-Index (ZRI) material using a phononic crystal (PC) composed of a square array of densely packed square iron rods in air. The dispersion relation exhibits a nearly flat band across the Brillouin zone at the reduced frequency f = 0.5443c/a, which is due to Fabry-Perot (FP) resonance. By using a retrieval method, we find that both the effective mass density and the reciprocal of the effective bulk modulus are close to zero at frequencies near the flat band. We also propose an equivalent tube network model to explain the mechanisms of the near ZRI effect. This FP-resonance-induced near ZRI material offers intriguing wave manipulation properties. We demonstrate both numerically and experimentally its ability to shield a scattering obstacle and guide acoustic waves through a bent structure.