The selection of low-radioactive construction materials is of the utmost importance for rare-event searches and thus critical to the XENONnT experiment. Results of an extensive radioassay program are reported, in which material samples have been screened with gamma-ray spectroscopy, mass spectrometry, and 222Rn emanation measurements. Furthermore, the cleanliness procedures applied to remove or mitigate surface contamination of detector materials are described. Screening results, used as inputs for a XENONnT Monte Carlo simulation, predict a reduction of materials background (∼17%) with respect to its predecessor XENON1T. Through radon emanation measurements, the expected 222Rn activity concentration in XENONnT is determined to be 4.2 (+0.5−0.7) μBq/kg, a factor three lower with respect to XENON1T. This radon concentration will be further suppressed by means of the novel radon distillation system.