The resonant substructures of B0→D¯0π+π- decays are studied with the Dalitz plot technique. In this study a data sample corresponding to an integrated luminosity of 3.0 fb-1 of pp collisions collected by the LHCb detector is used. The branching fraction of the B0→D¯0π+π- decay in the region m(D¯0π±)>2.1 GeV/c2 is measured to be (8.46±0.14±0.29±0.40)×10-4, where the first uncertainty is statistical, the second is systematic and the last arises from the normalization channel B0→D∗(2010)-π+. The π+π- 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(D¯0π-)≈2.8 GeV/c2 is confirmed and its spin-parity is determined for the first time as JP=3-. The branching fraction, mass and width of this structure are determined together with those of the D0∗(2400)- and D2∗(2460)- resonances. The branching fractions of other B0→D¯0h0 decay components with h0→π+π- are also reported. Many of these branching fraction measurements are the most precise to date. The first observation of the decays B0→D¯0f0(500), B0→D¯0f0(980), B0→D¯0ρ(1450), B0→D3∗(2760)-π+ and the first evidence of B0→D¯0f0(2020) are presented.