The coming flood of cosmic microwave background (CMB) polarization experiments, spurred by the recent detection of CMB polarization by the DASI and WMAP instruments, will be confronted by many new analysis tasks specific to polarization. For the analysis of CMB polarization data sets, the devil is truly in the details. With this in mind, we present details of the data analysis for the POLAR experiment, which recently led to the tightest upper limits on the polarization of the cosmic microwave background radiation at large angular scales. We discuss the data selection process, map-making algorithms, offset removal, and likelihood analysis which were used to find upper limits on the polarization. Stated using the modern convention for reporting CMB Stokes parameters, these limits are 5.0 μK on both E- and B-type polarization at 95% confidence. Finally, we discuss simulations used to test our analysis techniques and to probe the fundamental limitations of the experiment.