Three-dimensional (3D) printing technology has great potential for constructing structurally and functionally complex scaffold materials for tissue engineering. Bio-inks are a critical part of 3D printing for this purpose. In this study, based on dynamic hydrazone-crosslinked hyaluronic acid (HA-HYD) and photocrosslinked gelatin methacrylate (GelMA), a double-network (DN) hydrogel with significantly enhanced mechanical strength, self-healing, and shear-thinning properties was developed as a printable hydrogel bio-ink for extrusion-based 3D printing. Owing to shear thinning, the DN hydrogel bio-inks could be extruded to form uniform filaments, which were printed layer by layer to fabricate the scaffolds. The self-healing performance of the filaments and photocrosslinking of GelMA worked together to obtain an integrated and stable printed structure with high mechanical strength. The in vitro cytocompatibility assay showed that the DN hydrogel printed scaffolds supported the survival and proliferation of bone marrow mesenchymal stem cells. GelMA/HA-HYD DN hydrogel bio-inks with printability, good structural integrity, and biocompatibility are promising materials for 3D printing of tissue engineering scaffolds.