A search for exotic decays of the Higgs boson H into new scalar or pseudoscalar particles that subsequently decay into b-quarks is presented. The search considers ZH production with several decay scenarios for the Higgs boson; first to a pair of identical scalars, H→2a→4b, second to a pair of scalars with different masses (ma1<ma2), either directly, H→a1a2→4b, or via a longer decay chain, H→a1a2→3a1→6b. The analysis uses proton-proton collision data at s=13 TeV collected with the ATLAS detector at the Large Hadron Collider, corresponding to an integrated luminosity of 140 fb-1. No significant excess above the Standard Model prediction is observed. The search sets upper limits at 95% confidence level on the ratio of the Higgs boson production cross section to the SM prediction times the branching ratio of Higgs bosons decaying into 4b or 6b, between 4% and 25% for σ(ZH)/σSM(ZH)×B(H→2a→4b), between 24% and 38% for σ(ZH)/σSM(ZH)×B(H→a1a2→4b), and between 10% and 20% for σ(ZH)/σSM(ZH)×B(H→a1a2→3a1→6b), depending on the masses of the scalar particles.