The Auger Engineering Radio Array (AERA), part of the Pierre Auger Observatory, is currently the largest facility to measure radio emissions from ultra-high energy extensive air showers. It comprises 153 autonomous radio-detector stations, covering an area of 17 km², and measures radio waves in the frequency range from 30 to 80 MHz. An accurate description of the detector response is necessary to interpret the data collected by the stations correctly. Previously, this was achieved by measuring the analog chain in the laboratory and simulating and measuring the directional response of the antenna. In this work, we perform an absolute calibration using the continuously monitored sidereal modulation of the diffuse Galactic radio emission. The calibration is performed by comparing the average spectra recorded by the stations with a model of the full radio sky propagated through the system response, including the antenna, filters and amplifiers. We describe the method to determine the calibration constants for each antenna and present the corresponding results. Furthermore, the behavior of the calibration constants is studied as a function of time. There is no relevant aging effect over a timescale of a decade, which shows that radio detectors could help monitor possible aging effects of other detector systems during long-term operations, stressing their importance in determining an absolute energy scale.