Rubbers exhibit enormous extensibility up to several hundred per cent, compared with a few per cent for ordinary solids, and have the ability to recover their original shape and dimensions on release of stress. Rubber elasticity is a property of macromolecules that are either covalently cross-linked or connected in a network by physical associations such as small glassy or crystalline domains, ionic aggregates or multiple hydrogen bonds. Covalent cross-links or strong physical associations prevent flow and creep. Here we design and synthesize molecules that associate together to form both chains and cross-links via hydrogen bonds. The system shows recoverable extensibility up to several hundred per cent and little creep under load. In striking contrast to conventional cross-linked or thermoreversible rubbers made of macromolecules, these systems, when broken or cut, can be simply repaired by bringing together fractured surfaces to self-heal at room temperature. Repaired samples recuperate their enormous extensibility. The process of breaking and healing can be repeated many times. These materials can be easily processed, re-used and recycled. Their unique self-repairing properties, the simplicity of their synthesis, their availability from renewable resources and the low cost of raw ingredients (fatty acids and urea) bode well for future applications.