Photonic crystal fibers (PCFs) provide a versatile platform for various applications, thanks to the flexibility with which light guiding can be customized by modifying the fiber geometry. We realize a PCF with guided modes produced by photonic band structure topology rather than conventional mode-trapping mechanisms. The design, which is compatible with the stack-and-draw fabrication process, consists of a cross-sectional photonic topological crystalline insulator with a disclination. A bulk-defect correspondence produces degenerate topological modes, lying below the cladding light line. We use various theoretical methods to confirm their topological origins, including a spectral localizer that makes minimal assumptions about the band structure. Our experiments on the fabricated fiber show it transmitting visible to near-infrared light with low losses of 10 to 20 decibels per kilometer, which do not increase substantially when the fiber is bent. A comparable solid-core PCF of conventional design exhibits substantially higher bending losses. Optical fibers based on topological modes hold promise for improved performance and versatile functionalities.