We present new constructions of quantum codes of linear or close-to-linear distance and dimension with low-weight stabilizers. Only a few constructions of such codes were previously known, and were primarily based on a specific operation from homological algebra, namely the balanced product. In contrast, our constructions are based on a more basic and widely used product, namely the homological product (i.e. the tensor product of chain complexes). Our results help address the natural question: When do homological products preserve good code distance? Our first main result constructs asymptotically good $[[N,Θ(N),Θ(N)]]$ quantum codes with small polynomial stabilizer weight from homological products of codes with a property called product-expansion. This notion was recently introduced and used to bound the distance of balanced product quantum codes; we apply it instead to homological products. For every $ε>0$, our second main result constructs close-to-linear distance $[[N,N^{1-ε},N^{1-ε}]]$ (subsystem) quantum LDPC codes with constant stabilizer weight from iterated homological products of a constant-sized quantum locally testable code. The key insight here is that by using subsystem codes (but still with constant-weight stabilizers), we can circumvent a particular obstruction that limited the distance of many prior product code constructions to at most $\tilde{O}(\sqrt{N})$.