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