We demonstrate the first system for optical coherence tomography (OCT) that enables simultaneous measurement of quantitative birefringence and strain in biological samples using a common-path configuration. Owing to its superior phase stability, common-path polarization sensitive optical coherence tomography (CoPPSe-OCT) achieves a sub-nanometer displacement sensitivity of 0.52 nm at an SNR of 48 dB. We utilize CoPPSe-OCT to measure reflectance, birefringence, and strain for distinguishing burnt regions in a birefringent biological sample (chicken breast muscle).