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$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ of data recorded in the ATLAS detector at the LHC at centre-of-mass energy $s$ = 8 TeV. In the four-lepton invariant-mass range of 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.