We study the relationship between age, metallicity, and α-enhancement of FGK stars in the Galactic disk. The results are based upon the analysis of high-resolution UVES spectra from the Gaia-ESO large stellar survey. We explore the limitations of the observed dataset, i.e. the accuracy of stellar parameters and the selection effects that are caused by the photometric target preselection. We find that the colour and magnitude cuts in the survey suppress old metal-rich stars and young metal-poor stars. This suppression may be as high as 97% in some regions of the age-metallicity relationship. The dataset consists of 144 stars with a wide range of ages from 0.5 Gyr to 13.5 Gyr, Galactocentric distances from 6 kpcto 9.5 kpc, and vertical distances from the plane 0 < |Z| < 1.5 kpc. On this basis, we find that i) the observed age-metallicity relation is nearly flat in the range of ages between 0 Gyr and 8 Gyr; ii) at ages older than 9 Gyr, we see a decrease in [Fe/H] and a clear absence of metal-rich stars; this cannot be explained by the survey selection functions; iii) there is a significant scatter of [Fe/H] at any age; and iv) [Mg/Fe] increases with age, but the dispersion of [Mg/Fe] at ages >9 Gyr is not as small as advocated by some other studies. In agreement with earlier work, we find that radial abundance gradients change as a function of vertical distance from the plane. The [Mg/Fe] gradient steepens and becomes negative. In addition, we show that the inner disk is not only more α-rich compared to the outer disk, but also older, as traced independently by the ages and Mg abundances of stars. Based on observations made with the ESO/VLT, at Paranal Observatory, under programme 188.B-3002 (The Gaia-ESO Public Spectroscopic Survey).