The LUX-ZEPLIN (LZ) experiment will search for dark matter particle interactions with a detector containing a total of 10 tonnes of liquid xenon within a double-vessel cryostat. The large mass and proximity of the cryostat to the active detector volume demand the use of material with extremely low intrinsic radioactivity. We report on the radioassay campaign conducted to identify suitable metals, the determination of factors limiting radiopure production, and the selection of titanium for construction of the LZ cryostat and other detector components. This titanium has been measured with activities of $^{238}$U$_{e}$~$<$1.6~mBq/kg, $^{238}$U$_{l}$~$<$0.09~mBq/kg, $^{232}$Th$_{e}$~$=0.28\pm 0.03$~mBq/kg, $^{232}$Th$_{l}$~$=0.25\pm 0.02$~mBq/kg, $^{40}$K~$<$0.54~mBq/kg, and $^{60}$Co~$<$0.02~mBq/kg (68\% CL). Such low intrinsic activities, which are some of the lowest ever reported for titanium, enable its use for future dark matter and other rare event searches. Monte Carlo simulations have been performed to assess the expected background contribution from the LZ cryostat with this radioactivity. In 1,000 days of WIMP search exposure of a 5.6-tonne fiducial mass, the cryostat will contribute only a mean background of $0.160\pm0.001$(stat)$\pm0.030$(sys) counts. Comment: 13 pages, 3 figures, accepted for publication in Astroparticle Physics
