A search is presented for a high-mass Higgs boson in the $H\rightarrow\ell\ell\ell\ell$, $H\rightarrow\ell\ell\nu\nu$, $H\rightarrow\ell\ell qq$, and $H\rightarrow\nu\nu qq$ decay modes using the ATLAS detector at the CERN Large Hadron Collider. The search uses proton-proton collision data at a centre-of-mass energy of 8 TeV corresponding to an integrated luminosity of 20.3 fb$^{-1}$. The results of the search are interpreted in the scenario of a heavy Higgs boson with a width that is small compared with the experimental mass resolution. The Higgs boson mass range considered extends up to 1 TeV for all four decay modes and down to as low as 140 GeV, depending on the decay mode. No significant excess of events over the Standard Model prediction is found. A simultaneous fit to the four decay modes yields upper limits on the production cross-section of a heavy Higgs boson times the branching ratio to Z boson pairs. 95% confidence level upper limits range from 0.53 pb at $m_H=195$ GeV to 0.008 pb at $m_H=950$ GeV for the gluon-fusion production mode and from 0.31 pb at $m_H=195$ GeV to 0.009 pb at $m_H=950$ GeV for the vector-boson-fusion production mode. The results are also interpreted in the context of Type-I and Type-II two-Higgs-doublet models.