We present first measurable predictions for electromagnetic (two-photon) double scattering production of two positron-electron pairs in ultraperipheral heavy-ion collisions at LHC. Measureable cross sections are obtained with realistic cuts on electron/positron (pseudo)rapidities and transverse momenta for the ALICE and ATLAS or CMS experiments. The predictions for total and differential cross sections are presented. We show also two-dimensional distributions in rapidities of the opposite-sign (from the same or different subcollisions) and of the same-sign ($e^+ e^+$ or $e^- e^-$) electrons and in rapidity distance between them. Expected number of events are presented and discussed. Our calculations strongly suggest that relevant measurements with the help of ATLAS, CMS and ALICE detectors are possible in a near future.