We search for the decays B0→ρ0ρ0, B0→ρ0f0(980), and B0→f0(980)f0(980) in a sample of about 384×106 Υ(4S) →BB̄ decays collected with the BABAR detector at the PEP-II asymmetric-energy e+e- collider at Stanford Linear Accelerator Center. We find evidence for B0→ρ0ρ0 with 3.5σ significance and measure the branching fraction B=(1.07±0.33±0.19)×10-6 and longitudinal polarization fraction fL=0.87±0.13±0.04, where the first uncertainty is statistical, and the second is systematic. The uncertainty on the Cabibbo-Kobayashi-Maskawa quark-mixing matrix unitarity angle α due to penguin contributions in B→ρρ decays is 18° at the 1σ level. We also set upper limits on the B0→ρ0f0(980) and B0→f0(980)f0(980) decay rates. © 2007 The American Physical Society.