Vector beams, structured optical beams with nonuniform polarization distributions over the cross-section, have been recently found to be more beneficial than the scalar beams to many applications ranging from super-resolution imaging and creation of strongly localized spins or magnetic resonances to multimode classical and quantum communications. The distribution of vector beams over a strand of low-cost optical fiber would alleviate the space constraint in their use and radically broaden their applications. Here, we experimentally demonstrate that both the radially polarized and azimuthally polarized vector beams can be stably guided over a wide spectral range in a highly germanium-doped-core optical fiber even with a simple step-index profile. Our careful fiber characterization and full-vectorial theoretical analysis show that the heavy-doping-induced significant in-fiber stress that is usually developed during the fiber drawing process plays a crucial role in robust vector beam guidance. Our study provides entirely new perspectives of fiber design suitable for cost-effective multimode optical communications and highly efficient multimode nonlinear fiber optics.