Resonator fiber optic gyroscope (RFOG) performance has hitherto been limited by nonlinearity, modal impurity, and backscattering in the sensing fibers. The use of hollow-core fiber (HCF) effectively reduces nonlinearity, but the complex interplay among glass and air-guided modes in conventional HCF technologies can severely exacerbate RFOG instability. By employing high-performance nested anti-resonant nodeless fiber, we demonstrate long-term stability in a hollow-fiber RFOG of 0.05 deg/h, nearing the levels required for civil aircraft navigation. This represents a ${{3}} \times$ improvement over any prior hollow-core RFOG and a factor of ${{500}} \times$ over any prior result at integration times longer than 1 h.