The inclusive production of four isolated charged leptons in $pp$ collisions is analysed for the presence of hard double-parton scattering, using 20.2 fb$^{-1}$ of data recorded in the ATLAS detector at the LHC at centre-of-mass energy $\sqrt{s}$ = 8 TeV. In the four-lepton invariant-mass range of $80<m_{4\ell}<1000$ GeV, an artificial neural network is used to enhance the separation between single- and double-parton scattering based on the kinematics of the four leptons in the final state. An upper limit on the fraction of events originating from double-parton scattering is determined at 95% confidence level to be $f_{\mathrm{DPS}} = 0.042$, which results in an estimated lower limit on the effective cross section at 95% confidence level of $1.0$ mb.