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A measurement of $ZZ$ anomalous trilinear gauge couplings, $f_i^V$ where $i=4,5$ and $V=Z,\gamma$, in LHC proton-proton collisions at $\sqrt{s}=8$ TeV has been performed using data collected with the ATLAS detector in 2012, with electrons and muons in the final state. The dataset corresponds to an integrated luminosity of 20.3$\pm$0.6 fb$^{-1}$. In total, 272 candidate $ZZ$ events are observed where both lepton pairs have an invariant mass in the range 66-116 GeV with an expected number of events from SM $ZZ$ of 227.3$\pm0.3$ and a background expectation of 10.3$\pm2.0$. The sensitivity of a range of observables has been evaluated and the transverse momentum of the leading lepton is found to be the more sensitive observable and thus used for the aTGC measurement. Even though there is an excess in the total event count in data, the distribution of events in the observable does not favour a non-zero value of aTGCs since the excess is located at low transverse momentum while a small deficit is observed at high transverse momentum where the sensitivity is largest. A binned maximum likelihood fit finds that the preferred values of aTGCs are $f_i^V=0$. Different statistical methods for deriving confidence intervals and contours have been investigated and it is concluded that among the three methods tested which are the method of profile likelihood, a method here denoted the $p$-value method and the Neyman construction, the first two systematically over-constrain the parameters in scenarios where less events are observed than what is expected from the SM, while the Neyman construction by definition gives the correct coverage. For the present observation in data, the confidence intervals and contours follow this pattern. The confidence belt gives the following confidence intervals: $-0.0104 < f_4^\gamma < 0.0104$, $-0.0088 < f_4^Z < 0.0086$, $-0.0104 < f_5^\gamma < 0.0104$,$-0.0088 < f_5^Z < 0.0090$. Compared to previous results, the present analysis improves the former best limits set by the CMS experiment using an integrated luminosity of 5.0 fb$^{-1}$ by $\sim30\%$.