An analysis is presented of events containing jets including at least one b-tagged jet, sizeable missing transverse momentum, and at least two leptons including a pair of the same electric charge, with the scalar sum of the jet and lepton transverse momenta being large. A data sample with an integrated luminosity of 20.3 fb−1 of pp collisions at s√=8 TeV recorded by the ATLAS detector at the Large Hadron Collider is used. Standard Model processes rarely produce these final states, but there are several models of physics beyond the Standard Model that predict an enhanced rate of production of such events; the ones considered here are production of vector-like quarks, enhanced four-top-quark production, pair production of chiral b′-quarks, and production of two positively charged top quarks. Eleven signal regions are defined; subsets of these regions are combined when searching for each class of models. In the three signal regions primarily sensitive to positively charged top quark pair production, the data yield is consistent with the background expectation. There are more data events than expected from background in the set of eight signal regions defined for searching for vector-like quarks and chiral b′-quarks, but the significance of the discrepancy is less than two standard deviations. The discrepancy reaches 2.5 standard deviations in the set of five signal regions defined for searching for four-top-quark production. The results are used to set 95% CL limits on various models.