This paper presents an advanced two-stage nanocomposite membrane system designed to efficiently separate and capture methane (CH4) and carbon dioxide (CO2) from atmospheric air and water sources. The membrane system comprises a CO2-selective primary membrane and a CH4-selective secondary membrane, utilizing a hierarchical nanomaterials and polymers structure. The proposed system demonstrates unprecedented versatility, operating effectively across an extensive range of gas concentrations (>20% to <0.02%) and reducing CH4 levels from 100-500 ppm to 5-10 ppm in both aerobic and anaerobic conditions. Performance metrics specify CO2 permeances of 200-2000 GPU and CO2/N2 selectivities of 30-500 at 57 °C and 1 atm feed pressure, surpassing the Robeson upper bound for traditional polymer membranes. The CH4-selective membrane achieves 500-2000 GPU permeances with CH4/CO2 selectivities >50. Furthermore, experimental validation over 1000 hours of continuous operation demonstrated 92% methane capture efficiency under challenging conditions (55 tons/hour methane content at 30 °C). The system's energy consumption of 0.3 kWh/kg of CH4 captured underscores its efficiency compared to traditional methods. This innovative membrane technology offers a promising solution for addressing critical ecological and industrial challenges associated with greenhouse gas emissions in the 21st century.