In this study, a high-sensitivity, high-spatial-resolution distributed strain-sensing approach based on a poly(methyl methacrylate) chirped fiber Bragg grating (CFBG) is proposed and experimentally demonstrated. Linearly chirped FBGs in a polymer optical fiber provide an alternative to the silica fiber owing to the lower Young’s modulus, which can yield a higher stress sensitivity under the same external force. According to the spatial wavelength-encoded characteristic of the CFBG, a fully distributed strain measurement can be achieved by optical frequency-domain reflectometry. Through time-/space-resolved short-time Fourier transform, the applied force can be located by the beat frequency originated from the space-induced time delay and measured by the differential frequency offset originated from the strain-induced dispersion time delay. In a proof-of-concept experiment, a high spatial resolution of 1 mm over a gauge length of 40 mm and a strain resolution of 0.491 Hz/με were achieved.