  We introduce a new topological quantum code, the three-dimensional subsystem
toric code (3D STC), which is a generalization of the stabilizer toric code.
The 3D STC can be realized by measuring geometrically-local parity checks of
weight at most three on the cubic lattice with open boundary conditions. We
prove that single-shot quantum error correction (QEC) is possible with the 3D
STC, i.e., one round of local parity-check measurements suffices to perform
reliable QEC even in the presence of measurement errors. We also propose an
efficient single-shot QEC strategy for the 3D STC and investigate its
performance. In particular, we numerically estimate the resulting storage
threshold against independent bit-flip, phase-flip and measurement errors to be
$p_\text{STC} \approx 1.045\%$. Such a high threshold together with local
parity-check measurements of small weight make the 3D STC particularly
appealing for realizing fault-tolerant quantum computing.
