The production of $W$ boson pairs in proton-proton collisions at $\sqrt{s}=$ 8 TeV is studied using data corresponding to 20.3 fb$^{-1}$ of integrated luminosity collected by the ATLAS detector during 2012 at the CERN Large Hadron Collider. The $W$ bosons are reconstructed using their leptonic decays into electrons or muons and neutrinos. Events with reconstructed jets are not included in the candidate event sample. A total of 6636 $WW$ candidate events are observed. Measurements are performed in fiducial regions closely approximating the detector acceptance. The integrated measurement is corrected for all acceptance effects and for the $W$ branching fractions to leptons in order to obtain the total $WW$ production cross section, which is found to be 71.1$\pm1.1$(stat)$^{+5.7}_{-5.0}$(syst)$\pm1.4$ pb. This agrees with the next-to-next-to-leading-order Standard Model prediction of 63.2$^{+1.6}_{-1.4}$(scale)$\pm1.2$(PDF) pb. Fiducial differential cross sections are measured as a function of each of six kinematic variables. The distribution of the transverse momentum of the leading lepton is used to set limits on anomalous triple-gauge-boson couplings.