Abstract3D bioprinting is a promising technology to fabricate custom geometries for tissue engineering. However, most bioprintable hydrogels are weak and fragile, difficult to handle, and cannot mimic the mechanical behaviors of the native soft elastic tissues. A visible light crosslinked, single‐network, elastic, and biocompatible hydrogel system based on an acrylate triblock copolymer of poly(ethylene glycol) PEG and polycaprolactone (PCL) (PEG‐PCL‐DA) is developed. To enable its application in the bioprinting of soft tissues, the hydrogel system is modified on its printability and biodegradability. Furthermore, it is hypothesized that this elastic material can better transmit pulsatile forces to cells, leading to enhanced cellular response under mechanical stimulation. This central hypothesis is tested using vascular conduits with smooth muscle cells (SMCs) cultured under pulsatile forces in a custom‐made bioreactor. The results show that vascular conduits made of PEG‐PCL‐DA hydrogel faithfully recapitulate the rapid stretch and recoil under the pulsatile pressure from 1 to 3 Hz frequency, which induces a contractile SMC phenotype, consistently upregulating the core contractile transcription factors. In summary, this work demonstrates the potential of elastic hydrogel for 3D bioprinting of soft tissues by fine‐tuning the printability, and biodegradability, while possessing robust elastic properties suitable for manual handling and biomechanical stimulation.