Combination of searches for the resonant production of Higgs boson pairs is performed in the $b\bar{b}b\bar{b}$, $b\bar{b}\tau^+\tau^-$, and $b\bar{b}\gamma\gamma$ decay channels using up to 139~fb$^{-1}$ of proton-proton collision data at a center-of-mass energy of 13 TeV recorded by the ATLAS detector at the LHC. No significant excess over the expected background was observed, and upper limits were set at the 95\% confidence level on the production cross section of Higgs boson pairs from the decay of a narrow scalar resonance with masses ranging from 251 GeV to 5 TeV. The observed (expected) upper limits range from 0.96 to 600 fb (1.2 to 390 fb). These results are interpreted in the context of the Type-I Two-Higgs-Doublet Model (2HDM) and the Minimal Supersymmetric Standard Model (MSSM), tightening constraints on parameter spaces not previously excluded by other searches.