A search is presented for the pair production of heavy vectorlike quarks, $T\overline{T}$ or $B\overline{B}$, that decay into final states with jets and no reconstructed leptons. Jets in the final state are classified using a deep neural network as arising from hadronically decaying $W/Z$ bosons, Higgs bosons, top quarks, or background. The analysis uses data from the ATLAS experiment corresponding to $36.1\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of proton-proton collisions with a center-of-mass energy of $\sqrt{s}=13\text{ }\text{ }\mathrm{TeV}$ delivered by the Large Hadron Collider in 2015 and 2016. No significant deviation from the Standard Model expectation is observed. Results are interpreted assuming the vectorlike quarks decay into a Standard Model boson and a third-generation-quark, $T\ensuremath{\rightarrow}Wb,Ht,Zt$ or $B\ensuremath{\rightarrow}Wt,Hb,Zb$, for a variety of branching ratios. At 95% confidence level, the observed (expected) lower limit on the vectorlike $B$-quark mass for a weak-isospin doublet ($B$, $Y$) is 950 (890) GeV, and the lower limits on the masses for the pure decays $B\ensuremath{\rightarrow}Hb$ and $T\ensuremath{\rightarrow}Ht$, where these results are strongest, are 1010 (970) GeV and 1010 (1010) GeV, respectively.