A search is presented for new phenomena in events characterised by high jet multiplicity, no leptons (electrons or muons), and four or more jets originating from the fragmentation of $b$-quarks ($b$-jets). The search uses 139 fb$^{-1}$ of $\sqrt{s}$ = 13 TeV proton-proton collision data collected by the ATLAS experiment at the Large Hadron Collider during Run 2. The dominant Standard Model background originates from multijet production and is estimated using a data-driven technique based on an extrapolation from events with low $b$-jet multiplicity to the high $b$-jet multiplicities used in the search. No significant excess over the Standard Model expectation is observed and 95% confidence-level limits that constrain simplified models of R-parity-violating supersymmetry are determined. The exclusion limits reach 950 GeV in top-squark mass in the models considered.