Reducing the energy bandwidth of electrons in a lattice below the long-range Coulomb interaction energy promotes correlation effects. Moiré superlattices-which are created by stacking van der Waals heterostructures with a controlled twist angle1-3-enable the engineering of electron band structure. Exotic quantum phases can emerge in an engineered moiré flat band. The recent discovery of correlated insulator states, superconductivity and the quantum anomalous Hall effect in the flat band of magic-angle twisted bilayer graphene4-8 has sparked the exploration of correlated electron states in other moiré systems9-11. The electronic properties of van der Waals moiré superlattices can further be tuned by adjusting the interlayer coupling6 or the band structure of constituent layers9. Here, using van der Waals heterostructures of twisted double bilayer graphene (TDBG), we demonstrate a flat electron band that is tunable by perpendicular electric fields in a range of twist angles. Similarly to magic-angle twisted bilayer graphene, TDBG shows energy gaps at the half- and quarter-filled flat bands, indicating the emergence of correlated insulator states. We find that the gaps of these insulator states increase with in-plane magnetic field, suggesting a ferromagnetic order. On doping the half-filled insulator, a sudden drop in resistivity is observed with decreasing temperature. This critical behaviour is confined to a small area in the density-electric-field plane, and is attributed to a phase transition from a normal metal to a spin-polarized correlated state. The discovery of spin-polarized correlated states in electric-field-tunable TDBG provides a new route to engineering interaction-driven quantum phases.