Context. It is well known that the magnetic activity of solar-type stars decreases with age, but it is widely debated in the literature whether there is a smooth decline or if there is an early sharp drop until 1–2 Gyr that is followed by a relatively inactive constant phase. Aims. We revisited the activity-age relation using time-series observations of a large sample of solar twins whose precise isochronal ages and other important physical parameters have been determined. Methods. We measured the Ca II H and K activity indices using ≈9000 HARPS spectra of 82 solar twins. In addition, the average solar activity was calculated through asteroids and Moon reflection spectra using the same instrumentation. Thus, we transformed our activity indices into the S Mount Wilson scale (S_MW), recalibrated the Mount Wilson absolute flux and photospheric correction equations as a function of T_eff, and then computed an improved bolometric flux normalized activity index log R′_HK (T_eff) for the entire sample. Results. New relations between activity and the age of solar twins were derived by assessing the chromospheric age-dating limits using log R′_HK (T_eff). We measured an average solar activity of S_MW = 0.1712 ± 0.0017 during solar magnetic cycles 23–24 covered by HARPS observations, and we also inferred an average of S_MW = 0.1694 ± 0.0025 for cycles 10–24, anchored on a sunspot number correlation of S index versus. We also found a simple relation between the average and the dispersion of the activity levels of solar twins. This enabled us to predict the stellar variability effects on the age-activity diagram, and consequently, to estimate the chromospheric age uncertainties that are due to the same phenomena. The age-activity relation is still statistically significant up to ages around 6–7 Gyr, in agreement with previous works using open clusters and field stars with precise ages. Conclusions. Our research confirms that Ca II H & K lines remain a useful chromospheric evolution tracer until stars reach ages of at least 6–7 Gyr. We found evidence that for the most homogenous set of old stars, the chromospheric activity indices seem to continue to decrease after the solar age toward the end of the main sequence. Our results indicate that a significant part of the scatter observed in the age-activity relation of solar twins can be attributed to stellar cycle modulations effects. The Sun seems to have a normal activity level and variability for its age.