We study the interaction of an incident plane wave with a metamaterial periodic structure consisting of alternating layers of positive and negative refractive index with average zero refractive index. We show that the existence of very narrow resonance peaks for which giant absorption - 50% at layer thickness of 1% of the incident wavelength - is exhibited. Maximum absorption is obtained at a specific layer thickness satisfying the critical coupling condition. This phenomenon is explained by the Rayleigh anomaly and by the excitation of Fabry Perot modes in the periodic layer. In addition, we investigate the modes supported by the structures for several limiting cases, and show that zero phase accumulation in the periodic metamaterial is obtained at resonance.