  We determine the light baryon spectrum on ensembles generated by the
Coordinated Lattice Simulations (CLS) effort, employing $N_f=2+1$ flavours of
non-perturbatively improved Wilson fermions. The hadron masses are interpolated
and extrapolated within the quark mass plane, utilizing three distinct
trajectories, two of which intersect close to the physical quark mass point and
the third one approaching the SU(3) chiral limit. The results are extrapolated
to the continuum limit, utilizing six different lattice spacings ranging from
$a\approx 0.10\,$fm down to below $0.04\,$fm. The light pion mass varies from
$M_{\pi}\approx 429\,$MeV down to $127\,$MeV. In general, the spatial extent is
kept larger than four times the inverse pion mass and larger than $2.3\,$fm,
with additional small and large volume ensembles to investigate finite size
effects. We determine the Wilson flow scales $\sqrt{t_{0,{\rm
ph}}}=0.1449^{(7)}_{(9)}\,$fm and $t_0^*\approx t_{0,{\rm ph}}$ from the octet
cascade ($\Xi$ baryon). Determining the light baryon spectrum in the continuum
limit, we find the nucleon mass $m_N=941.7^{(6.5)}_{(7.6)}\,$MeV and the other
stable baryon masses to agree with their experimental values within sub-percent
level uncertainties. Moreover, we determine SU(3) and SU(2) chiral perturbation
theory low energy constants, including the octet and the $\Omega$ baryon
sigma~terms $\sigma_{\pi N}=43.9(4.7)\,$MeV,
$\sigma_{\pi\Lambda}=28.2^{(4.3)}_{(5.4)}\,$MeV,
$\sigma_{\pi\Sigma}=25.9^{(3.8)}_{(6.1)}\,$MeV,
$\sigma_{\pi\Xi}=11.2^{(4.5)}_{(6.4)}\,$MeV and
$\sigma_{\pi\Omega}=6.9^{(5.3)}_{(4.3)}\,$MeV, as well as various parameters,
renormalization factors and improvement coefficients that are relevant for
simulations with our lattice action.
