We introduce the THE THREE HUNDRED project, an endeavour to model 324 large galaxy clusters with full-physics hydrodynamical re-simulations. Here we present the data set and study the differences to observations for fundamental galaxy cluster properties and scaling relations. We find that the modelled galaxy clusters are generally in reasonable agreement with observations with respect to baryonic fractions and gas scaling relations at redshift z = 0. However, there are still some (model-dependent) differences, such as central galaxies being too massive, and galaxy colours (g - r) being bluer (about 0.2 dex lower at the peak position) than in observations. The agreement in gas scaling relations down to 10^{13} h^{-1} M_{⊙} between the simulations indicates that particulars of the sub-grid modelling of the baryonic physics only has a weak influence on these relations. We also include - where appropriate - a comparison to three semi-analytical galaxy formation models as applied to the same underlying dark-matter-only simulation. All simulations and derived data products are publicly available.
