This Letter describes a model-agnostic search for pairs of jets (dijets) produced by resonant and non-resonant phenomena beyond the Standard Model in 3.6 fb-1 of proton-proton collisions with a centre-of-mass energy of s=13 TeV recorded by the ATLAS detector at the Large Hadron Collider. The distribution of the invariant mass of the two leading jets is examined for local excesses above a data-derived estimate of the smoothly falling prediction of the Standard Model. The data are also compared to a Monte Carlo simulation of Standard Model angular distributions derived from the rapidity of the two jets. No evidence of anomalous phenomena is observed in the data, which are used to exclude, at 95% CL, quantum black holes with threshold masses below 8.3 TeV, 8.1 TeV, or 5.1 TeV in three different benchmark scenarios; resonance masses below 5.2 TeV for excited quarks, 2.6 TeV in a W' model, a range of masses starting from mZ'=1.5 TeV and couplings from gq=0.2 in a Z' model; and contact interactions with a compositeness scale below 12.0 TeV and 17.5 TeV respectively for destructive and constructive interference between the new interaction and QCD processes. These results significantly extend the ATLAS limits obtained from 8 TeV data. Gaussian-shaped contributions to the mass distribution are also excluded if the effective cross-section exceeds values ranging from approximately 50-300 fb for masses below 2 TeV to 2-20 fb for masses above 4 TeV.