The resonant substructures of ${B}^{0}\ensuremath{\rightarrow}{\overline{D}}^{0}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ decays are studied with the Dalitz plot technique. In this study a data sample corresponding to an integrated luminosity of $3.0\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of $pp$ collisions collected by the LHCb detector is used. The branching fraction of the ${B}^{0}\ensuremath{\rightarrow}{\overline{D}}^{0}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ decay in the region $m({\overline{D}}^{0}{\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}})>2.1\text{ }\text{ }\mathrm{GeV}/{c}^{2}$ is measured to be $(8.46\ifmmode\pm\else\textpm\fi{}0.14\ifmmode\pm\else\textpm\fi{}0.29\ifmmode\pm\else\textpm\fi{}0.40)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$, where the first uncertainty is statistical, the second is systematic and the last arises from the normalization channel ${B}^{0}\ensuremath{\rightarrow}{D}^{*}(2010{)}^{\ensuremath{-}}{\ensuremath{\pi}}^{+}$. The ${\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ S-wave components are modeled with the isobar and K-matrix formalisms. Results of the Dalitz plot analyses using both models are presented. A resonant structure at $m({\overline{D}}^{0}{\ensuremath{\pi}}^{\ensuremath{-}})\ensuremath{\approx}2.8\text{ }\text{ }\mathrm{GeV}/{c}^{2}$ is confirmed and its spin-parity is determined for the first time as ${J}^{P}={3}^{\ensuremath{-}}$. The branching fraction, mass and width of this structure are determined together with those of the ${D}_{0}^{*}(2400{)}^{\ensuremath{-}}$ and ${D}_{2}^{*}(2460{)}^{\ensuremath{-}}$ resonances. The branching fractions of other ${B}^{0}\ensuremath{\rightarrow}{\overline{D}}^{0}{h}^{0}$ decay components with ${h}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ are also reported. Many of these branching fraction measurements are the most precise to date. The first observation of the decays ${B}^{0}\ensuremath{\rightarrow}{\overline{D}}^{0}{f}_{0}(500)$, ${B}^{0}\ensuremath{\rightarrow}{\overline{D}}^{0}{f}_{0}(980)$, ${B}^{0}\ensuremath{\rightarrow}{\overline{D}}^{0}\ensuremath{\rho}(1450)$, ${B}^{0}\ensuremath{\rightarrow}{D}_{3}^{*}(2760{)}^{\ensuremath{-}}{\ensuremath{\pi}}^{+}$ and the first evidence of ${B}^{0}\ensuremath{\rightarrow}{\overline{D}}^{0}{f}_{0}(2020)$ are presented.