Developing closed-loop recyclable thermosets and understanding their structure-property relationships are essential steps in advancing a circular materials economy. Here, we present a vinylogous urethane (VU) thermoset with closed-loop recyclability, synthesized through the reaction of polytetrahydrofuran bisacetoacetate (aPTHF) and tris(2-aminoethyl)amine (TREN). These VU polymers exhibit high elasticity, with only a 3-9% residual strain observed after cyclic tensile testing at a maximum strain of 100%, depending on the molecular weight of aPTHF and network cross-link density. The two structural parameters also allow modulation of the mechanical and stress-relaxation properties of VU elastomers. To investigate the hydrolysis of the VU linkages within the hydrophobic aPTHF matrix, we employed a heterogeneous system using a biphasic mixture of HCl and CDCl3. Our findings show that the hydrophobic VUs remain stable in pure water but can be dissociated under acidic conditions, with the dissociation rate accelerated at higher temperatures and/or in the presence of higher HCl concentrations. These detailed investigations indicate the potential of VU elastomers as sustainable substrates for wearable sensors. We therefore conduct a case study of synthesizing a strain sensor through the incorporation of multiwalled carbon nanotubes (MCNs) into the VU elastomer matrix. The sensor can robustly detect various movements. Moreover, acidic treatment of both the neat polymer and the sensor composite using a HCl and diethyl ether solvent mixture allows for the excellent recovery of aPTHF (>90%) and TREN (86%), without discernible damage to the MCNs reclaimed from the latter.