We report a fiber-based lensless holographic imaging system to realize a single-shot measurement of two dimensional (2-D) Jones matrix parameters of polarization-sensitive materials. In this system, a multi-source lensless off-axis Fresnel holographic recording geometry is adopted, and two optical fiber splitters are used to generate the multiple reference and illumination beams required for recording a four-channel angular-multiplexing polarization hologram (AMPH). Using this system and the method described in this paper, spatially resolved Jones matrix parameters of a polarization-sensitive material can be retrieved from one single-shot AMPH. We demonstrate the feasibility of the method by extracting a 2-D Jones matrix of a composite polarizer. Applications of the method to measure the Jones matrix maps of a stressed polymethyl methacrylate sample and a mica fragment are also presented. Benefit from the fiber-based and lensless off-axis holographic design, the system possesses a quite compact configuration, which provides a feasible approach for development of an integrated and portable system to measure Jones matrix parameters of polarization-sensitive materials.