The ${\mathit{ppK}}^{\ensuremath{-}}$ system, as a prototype for possible quasibound $\overline{K}$ nuclei, is investigated using a variational approach. Several versions of energy-dependent effective $\overline{K}N$ interactions derived from chiral SU(3) dynamics are employed as input, together with a realistic $\mathit{NN}$ potential (Av18). Taking into account theoretical uncertainties in the extrapolations below the $\overline{K}N$ threshold, we find that the antikaonic dibaryon ${\mathit{ppK}}^{\ensuremath{-}}$ is not deeply bound. With the driving $s$-wave $\overline{K}N$ interaction the resulting total binding energy is $B({\mathit{ppK}}^{\ensuremath{-}})=20\ifmmode\pm\else\textpm\fi{}3$ MeV and the mesonic decay width involving $\overline{K}N\ensuremath{\rightarrow}\ensuremath{\pi}Y$ is expected to be in the range 40--70 MeV. Properties of this quasibound ${\mathit{ppK}}^{\ensuremath{-}}$ system (such as density distributions of nucleons and antikaon) are discussed. The $\ensuremath{\Lambda}(1405)$, as an $I=0$ quasibound state of $\overline{K}$ and a nucleon, appears to survive in the ${\mathit{ppK}}^{\ensuremath{-}}$ cluster. Estimates are given for the influence of $p$-wave $\overline{K}N$ interactions and for the width from two-nucleon absorption $(\overline{K}\mathit{NN}\ensuremath{\rightarrow}\mathit{YN})$ processes. With inclusion of these effects and dispersive corrections from absorption, the ${\mathit{ppK}}^{\ensuremath{-}}$ binding energy is expected to be in the range 20--40 MeV, whereas the total decay width can reach 100 MeV but with large theoretical uncertainties.