Measurements of the $ZZ$ and $WW$ final states in the mass range above the $2m_Z$ and $2m_W$ thresholds provide a unique opportunity to measure the off-shell coupling strength of the Higgs boson. This paper presents constraints on the off-shell Higgs boson event yields normalised to the Standard Model prediction (signal strength) in the $ZZ \to 4\ell$, $ZZ\to2\ell2\nu$ and $WW\to e\nu\mu\nu$ final states. The result is based on $pp$ collision data collected by the ATLAS experiment at the LHC, corresponding to an integrated luminosity of $20.3$ fb$^\text{-1}$ at a collision energy of $\sqrt{s}$=8 TeV. Using the $CL_s$ method, the observed 95% confidence level (CL) upper limit on the off-shell signal strength is in the range 5.1-8.6, with an expected range of 6.7-11.0. In each case the range is determined by varying the unknown $gg\to ZZ$ and $gg\to WW$ background K-factor from higher-order QCD corrections between half and twice the value of the known signal K-factor. Assuming the relevant Higgs boson couplings are independent of the energy scale of the Higgs production, a combination with the on-shell measurements yields an observed (expected) 95\% CL upper limit on $\Gamma_H/\Gamma_H^{\mathrm{SM}}$ in the range 4.5- -7.5 (6.5- -11.2) using the same variations of the background K-factor. Assuming that the unknown $gg\rightarrow VV$ background K-factor is equal to the signal K-factor,this translates into an observed (expected) 95% CL upper limit on the Higgs boson total width of 22.7 (33.0) MeV.