A study of inclusive four-jet production in proton-proton collisions at a center-of-mass energy of 13 TeV is presented. The transverse momentum of jets within abs(eta) < 4.7 reaches down to 35, 30, 25, and 20 GeV for the first-, second-, third-, and fourth-leading jet, respectively. Differential cross sections are measured as functions of the jet transverse momentum and pseudorapidity, and several other observables that describe the angular correlations between the jets. It is found that the measured distributions show sensitivity to different aspects of the underlying event, parton shower, and matrix element calculations. In particular, the interplay between decorrelations caused by parton shower and double-parton scattering contributions is shown to be important. The double-parton scattering contribution is extracted by means of a template fit of the distributions for single-parton scattering obtained from Monte Carlo event generators and a double-parton scattering distribution constructed from inclusive single-jet events in data. Values of the effective cross section are calculated and discussed in view of previous measurements and of their dependence on the models used to describe the single-parton scattering background.