Biodegradable polymers are an active area of investigation, particularly ones that can be produced from sustainable, biobased monomers, such as copolymers of poly(butylene succinate) (PBS). In this study, we examine the enzymatic degradation of poly(butylene succinate-dilinoleic succinate) (PBS-DLS) copolymers obtained by “green” enzymatic synthesis using lipase B from Candida antarctica (CALB). The copolymers differed in their hard to soft segments ratio, from 70:30 to 50:50 wt %. Enzymatic degradation was carried out on electrospun membranes (scaffolds) and compression-moulded films using lipase from Pseudomomas cepacia. Poly(ε-caprolactone) (PCL) was used as a reference aliphatic polyester. The degradation process was monitored gravimetrically via water uptake and mass loss. After 24 days, approx. 40% mass loss was observed for fibrous materials prepared from the PBS-DLS 70:30 copolymer, as compared to approx. 10% mass loss for PBS-DLS 50:50. Infrared spectroscopy (FTIR) and size exclusion chromatography (SEC) analysis were used to examine changes in chemical structure. Differential scanning calorimetry (DSC) and scanning light microscopy (LSM) revealed changes in degree of crystallinity, and changes in surface morphology, consistent with a surface erosion mechanism. We conclude that the obtained copolymers are suitable for tissue engineering applications thanks to tuneable degradation and lack of acidification during breakdown.