The first measurements of diboson production cross sections in proton-proton interactions at a center-of-mass energy of 5.02 TeV are reported. They are based on data collected with the CMS detector at the LHC, corresponding to an integrated luminosity of $302\text{ }\text{ }{\mathrm{pb}}^{\ensuremath{-}1}$. Events with two, three, or four charged light leptons (electrons or muons) in the final state are analyzed. The $WW$, $WZ$, and $ZZ$ total cross sections are measured as ${\ensuremath{\sigma}}_{WW}=37.{0}_{\ensuremath{-}5.2}^{+5.5}(\mathrm{stat}{)}_{\ensuremath{-}2.6}^{+2.7}(\mathrm{syst})\text{ }\text{ }\mathrm{pb}$, ${\ensuremath{\sigma}}_{WZ}=6.{4}_{\ensuremath{-}2.1}^{+2.5}(\mathrm{stat}{)}_{\ensuremath{-}0.3}^{+0.5}(\mathrm{syst})\text{ }\text{ }\mathrm{pb}$, and ${\ensuremath{\sigma}}_{ZZ}=5.{3}_{\ensuremath{-}2.1}^{+2.5}(\mathrm{stat}{)}_{\ensuremath{-}0.4}^{+0.5}(\mathrm{syst})\text{ }\text{ }\mathrm{pb}$. All measurements are in good agreement with theoretical calculations at combined next-to-next-to-leading order quantum chromodynamics and next-to-leading order electroweak accuracy.