In the biomedical field, degradable chemically crosslinked elastomers are interesting materials for tissue engineering applications, since they present rubber-like mechanical properties matching those of soft tissues and are able to preserve their three-dimensional (3D) structure over degradation. Their use in biomedical applications requires surgical handling and implantation that can be a source of accidental damages responsible for the loss of properties. Therefore, their inability to be healed after damage or breaking can be a major drawback. In this work, biodegradable dual-crosslinked networks that exhibit fast and efficient self-healing properties at 37 °C are designed. Self-healable dual-crosslinked (chemically and physically) elastomeric networks are prepared by two methods. The first approach is based on the mix of hydrophobic poly(ethylene glycol)-poly(lactic acid) (PEG-PLA) star-shaped copolymers functionalized with either catechol or methacrylate moieties. In the second approach, hydrophobic bifunctional PEG-PLA star-shaped copolymers with both catechol and methacrylate on their structure are used. In the two systems, the supramolecular network is responsible for the self-healing properties, thanks to the dynamic dissociation/reassociation of the numerous hydrogen bonds between the catechol groups, whereas the covalent network ensures mechanical properties similar to pure methacrylate networks. The self-healable materials display mechanical properties that are compatible with soft tissues and exhibit linear degradation because of the chemical cross-links. The performances of networks from mixed copolymers versus bifunctional copolymers are compared and demonstrate the superiority of the latter. The biocompatibility of the materials is also demonstrated, confirming the potential of these degradable and self-healable elastomeric networks to be used for the design of temporary medical devices.