We design an all-DNA system that behaves like vitrimers, innovative plastics with self-healing and stress-releasing properties. The DNA sequences are engineered to self-assemble first into tetra- and bifunctional units which, upon further cooling, bind to each other forming a fully bonded network gel. An innovative design of the binding regions of the DNA sequences, exploiting a double toehold-mediated strand displacement, generates a network gel which is able to reshuffle its bonds, retaining at all times full bonding. As in vitrimers, the rate of bond switching can be controlled via a thermally activated catalyst, which in the present design is very short DNA strands.