The Close AGN Reference Survey (CARS): IFU survey data and the BH mass dependence of long-term AGN variability

[Abridged] AGN are thought to be intimately connected with their host galaxies through feeding and feedback processes. A spatially resolved multiwavelength survey is required to map the interaction of AGN with their host galaxies on different spatial scales and different phases of the ISM. The goal of CARS is to obtain the necessary spatially resolved multiwavelength observations for an unbiased sample of local unobscured luminous AGN. We present the overall CARS survey design and the associated wide-field optical IFU spectroscopy for all 41 CARS targets at z<0.06 randomly selected from the Hamburg/ESO survey of luminous unobscured AGN. This data set provides the backbone of CARS and allows us to characterize host galaxy morphologies, AGN parameters, precise systemic redshifts, and ionized gas distributions including excitation conditions, kinematics, and metallicities in unprecedented detail. We focus our study on the size of the ENLR which has been traditionally connected to AGN luminosity. Given the large scatter in the ENLR size-luminosity relation, we performed a large parameter search to identify potentially more fundamental relations. Remarkably, we identified the strongest correlation between the maximum projected ENLR size and the black hole mass, consistent with an $R_\mathrm{ENLR,max}\sim M_\mathrm{BH}^{0.5}$ relationship. We interpret the maximum ENLR size as a timescale indicator of a single BH radiative-efficient accretion episode for which we inferred log(t_AGN) = (0.45+- 0.08)log(M_BH)+1.78 using forward modeling. The extrapolation of our inferred relation toward higher BH masses is consistent with an independent lifetime estimate from the HeII proximity zones around luminous AGN at z~3. While our proposed link between the BH mass and AGN lifetime might be a secondary correlation itself or impacted by unknown biases, it has a few relevant implications if confirmed.