We present a study of four muon, four electron, and two muon two electron events using 1 fb^(-1) of data collected with the D0 detector at the Fermilab Tevatron p-barp Collider at sqrt(s) = 1.96 TeV. Requiring the lepton pair masses to be greater than 30 GeV, we observe one event, consistent with the expected background of 0.13 +- 0.03 events and with the predicted standard model ZZ and Z\gamma^* production of 1.71 +-0.15 events. We set an upper limit on the ZZ and Z\gamma^* cross section of 4.4 pb at the 95% C.L. We also derive limits on anomalous neutral trilinear ZZZ and ZZ\gamma^* gauge couplings. The one-parameter 95%$ C.L. coupling limits with a form factor scale Lambda = 1.2 TeV are -0.28