Aims. RefPlanets is a guaranteed time observation programme that uses the Zurich IMaging POLarimeter (ZIMPOL) of Spectro-Polarimetric High-contrast Exoplanet REsearch instrument at the Very Large Telescope to perform a blind search for exoplanets in wavelengths from 600 to 900 nm. The goals of this study are the characterisation of the unprecedented high polarimetic contrast and polarimetric precision capabilities of ZIMPOL for bright targets, the search for polarised 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. Methods. For our observations of α Cen A and B, Sirius A, Altair, ɛ Eri and τ Ceti we used the polarimetricdifferential 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 as a final layer of post-processing to further improve the contrast limits of our data at these separations. Results. For good observing conditions we achieve polarimetric contrast limits of 15.0–16.3 mag at the effective inner working angle of ~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′′) display a significant dependence on 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 polarised light sources in the data obtained so far.