Searches for exclusively produced $W$ boson pairs in the process $pp(\ensuremath{\gamma}\ensuremath{\gamma})\ensuremath{\rightarrow}p{W}^{+}{W}^{\ensuremath{-}}p$ and an exclusively produced Higgs boson in the process $pp(gg)\ensuremath{\rightarrow}pHp$ have been performed using ${e}^{\ifmmode\pm\else\textpm\fi{}}{\ensuremath{\mu}}^{\ensuremath{\mp}}$ final states. These measurements use $20.2\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of $pp$ collisions collected by the ATLAS experiment at a center-of-mass energy $\sqrt{s}=8\text{ }\text{ }\mathrm{TeV}$ at the LHC. Exclusive production of ${W}^{+}{W}^{\ensuremath{-}}$ consistent with the Standard Model prediction is found with $3.0\ensuremath{\sigma}$ significance. The exclusive ${W}^{+}{W}^{\ensuremath{-}}$ production cross section is determined to be $\ensuremath{\sigma}(\ensuremath{\gamma}\ensuremath{\gamma}\ensuremath{\rightarrow}{W}^{+}{W}^{\ensuremath{-}}\ensuremath{\rightarrow}{e}^{\ifmmode\pm\else\textpm\fi{}}{\ensuremath{\mu}}^{\ensuremath{\mp}}X)=6.9\ifmmode\pm\else\textpm\fi{}2.2(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}1.4(\mathrm{sys})\text{ }\text{ }\mathrm{fb}$, in agreement with the Standard Model prediction. Limits on anomalous quartic gauge couplings are set at 95% confidence level as $\ensuremath{-}1.7\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}<{a}_{0}^{W}/{\mathrm{\ensuremath{\Lambda}}}^{2}<1.7\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}\text{ }\text{ }{\mathrm{GeV}}^{\ensuremath{-}2}$ and $\ensuremath{-}6.4\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}<{a}_{C}^{W}/{\mathrm{\ensuremath{\Lambda}}}^{2}<6.3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}\text{ }\text{ }{\mathrm{GeV}}^{\ensuremath{-}2}$. A 95% confidence-level upper limit on the total production cross section for an exclusive Higgs boson is set to 1.2 pb.