  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 $\sqrt{s} = 8$ 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
$b^\prime$-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 $b^\prime$-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.
