We demonstrate direct evidence for the first realization of atomically smooth sapphire crystalline fiber cores with a surface variation of only ~1.9 Å. The hybrid glass-clad crystalline cores were grown by a laser-based fiber drawing technique. Because of the improvement in crystal fiber quality, we were able, for the first time, to comprehensively and quantitatively elucidate the correlation between fiber nanostructure and optical loss. We also experimentally demonstrated that high-temperature treatment has a significant impact on defect relaxation and promotes excellent crystallinity, and hence enables low-loss optical wave guiding. The experimentally measured propagation losses in the order of 0.01-0.1 dB/cm are the lowest ever reported among conventional Ti:sapphire channel waveguides and ultrafast-laser-inscribed waveguides, and agree well with the theory. Through experiments and numerical calculation, we have demonstrated that low threshold and high efficiency of Ti:sapphire crystal fiber lasers are possible with the atomic-level roughness, low-loss propagation, and high crystallinity of the Ti:sapphire crystalline core.