A search for heavy Majorana neutrinos in events containing a pair of high-$p_{\mathrm{T}}$ leptons of the same charge and high-$p_{\mathrm{T}}$ jets is presented. The search uses $20.3 \mathrm{fb}^{-1}$ of $pp$ collision data collected with the ATLAS detector at the CERN Large Hadron Collider with a centre-of-mass energy of $\sqrt{s} = 8$ TeV. The data are found to be consistent with the background-only hypothesis based on the Standard Model expectation. In the context of a Type-I seesaw mechanism, limits are set on the production cross-section times branching ratio for production of heavy Majorana neutrinos in the mass range between 100 and 500 GeV. The limits are subsequently interpreted as limits on the mixing between the heavy Majorana neutrinos and the Standard Model neutrinos. In the context of a left-right symmetric model, limits on the production cross-section times branching ratio are set with respect to the masses of heavy Majorana neutrinos and heavy gauge bosons $W_{\mathrm{R}}$ and $Z'$.