We experimentally realize an optical fiber ring resonator that includes a tapered section with subwavelength-diameter waist. In this section, the guided light exhibits a significant evanescent field which allows for efficient interfacing with optical emitters. A commercial tunable fiber beam splitter provides simple and robust coupling to the resonator. Key parameters of the resonator such as its out-coupling rate, free spectral range, and birefringence can be adjusted. Thanks to the low taper- and coupling-losses, the resonator exhibits an unloaded finesse of F=75+/-1, sufficient for reaching the regime of strong coupling for emitters placed in the evanescent field. The system is ideally suited for trapping ensembles of laser-cooled atoms along the nanofiber section. Based on measured parameters, we estimate that the system can serve as a platform for optical multimode strong coupling experiments. Finally, we discuss the possibilities of using the resonator for applications based on chiral quantum optics.