Context.Achernar, the closest and brightest classical Be star, presents rotational flattening, gravity darkening, occasional emission lines due to a gaseous disk, and an extended polar wind. It is also a member of a close binary system with an early A-type dwarf companion.Aims.We aim to determine the orbital parameters of the Achernar system and to estimate the physical properties of the components.Methods.We monitored the relative position of Achernar B using a broad range of high angular resolution instruments of the VLT/VLTI (VISIR, NACO, SPHERE, AMBER, PIONIER, GRAVITY, and MATISSE) over a period of 13 years (2006−2019). These astrometric observations are complemented with a series of ≈750 optical spectra for the period from 2003 to 2016.Results.We determine that Achernar B orbits the primary Be star on a seven-year period, eccentric orbit (e = 0.7258 ± 0.0015) which brings the two stars within 2 au at periastron. The mass of the Be star is found to bem_A = 6.0 ± 0.6 M_⊙for a secondary mass ofm_B = 2.0 ± 0.1 M_⊙(the latter was estimated from modeling). We find a good agreement of the parameters of Achernar A with the evolutionary model of a critically rotating star of 6.4 M_⊙at an age of 63 Ma. The equatorial plane of the Be star and the orbital plane of the companion exhibit a relative inclination of 30°. We also identify a resolved comoving low-mass star, which leads us to propose that Achernar is a member of the Tucana-Horologium moving group.Conclusions.The proximity of Achernar makes this star a precious benchmark for stellar evolution models of fast rotators and intermediate mass binaries. Achernar A is presently in a short-lived phase of its evolution following the turn-off, during which its geometrical flattening ratio is the most extreme. Considering the orbital parameters, no significant interaction occurred between the two components, demonstrating that Be stars may form through a direct, single-star evolution path without mass transfer. Since component A will enter the instability strip in a few hundred thousand years, Achernar appears to be a promising progenitor of the Cepheid binary systems.