RefPlanets is a guaranteed time observation (GTO) programme that uses the Zurich IMaging POLarimeter (ZIMPOL) of SPHERE/VLT for a blind search for exoplanets in wavelengths from 600-900 nm. The goals of this study are the characterization of the unprecedented high polarimetic contrast and polarimetric precision capabilities of ZIMPOL for bright targets, the search for polarized reflected light around some of the closest bright stars to the Sun and potentially the direct detection of an evolved cold exoplanet for the first time. For our observations of Alpha Cen A and B, Sirius A, Altair, Eps Eri and Tau Ceti we used the polarimetric differential imaging (PDI) mode of ZIMPOL which removes the speckle noise down to the photon noise limit for angular separations >0.6. We describe some of the instrumental effects that dominate the noise for smaller separations and explain how to remove these additional noise effects in post-processing. We then combine PDI with angular differential imaging (ADI) as a final layer of post-processing to further improve the contrast limits of our data at these separations. For good observing conditions we achieve polarimetric contrast limits of 15.0-16.3 mag at the effective inner working angle of about 0.13, 16.3-18.3 mag at 0.5 and 18.8-20.4 mag at 1.5. The contrast limits closer in (<0.6) depend significantly on the observing conditions, while in the photon noise dominated regime (>0.6), the limits mainly depend on the brightness of the star and the total integration time. We compare our results with contrast limits from other surveys and review the exoplanet detection limits obtained with different detection methods. For all our targets we achieve unprecedented contrast limits. Despite the high polarimetric contrasts we are not able to find any additional companions or extended polarized light sources in the data that has been taken so far.