Context. Dome C in Antarctica is a promising site for photometric observations thanks to the continuous night during the Antarctic winter and favorable weather conditions. Aims.We developed instruments to assess the quality of this site for photometry in the visible and to detect and characterize variable objects through the Antarctic Search for Transiting ExoPlanets (ASTEP) project. Methods: Here, we present the full analysis of four winters of data collected with ASTEP South, a 10 cm refractor pointing continuously toward the celestial south pole. We improved the instrument over the years and developed specific data reduction methods. Results: We achieved nearly continuous observations over the winters. We measure an average sky background of 20 mag arcsec-2 in the 579-642 nm bandpass. We built the lightcurves of 6000 stars and developed a model to infer the photometric quality of Dome C from the lightcurves themselves. The weather is photometric 67.1 ± 4.2% of the time and veiled 21.8 ± 2.0% of the time. The remaining time corresponds to poor quality data or winter storms. We analyzed the lightcurves of σ Oct and HD 184465 and find that the amplitude of their main frequency varies by a factor of 3.5 and 6.7 over the four years, respectively. We also identify 34 new variable stars and eight new eclipsing binaries with periods ranging from 0.17 to 81 days. Conclusion. The phase coverage that we achieved with ASTEP South is exceptional for a ground-based instrument and the data quality enables the detection and study of variable objects. These results demonstrate the high quality of Dome C for photometry in the visible and for time series observations in general. Lightcurves are only available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (ftp://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/619/A116