Ultrathin electrospun fibers of pristine biopolyesters, poly(3-hydroxybutyrate) (PHB) and polylactic acid (PLA), as well as their blends, have been obtained and then explored after exposure to hydrolytic (phosphate buffer) and oxidative (ozone) media. All the fibers were obtained from a co-solvent, chloroform, by solution-mode electrospinning. The structure, morphology, and segmental dynamic behavior of the fibers have been determined by optical microscopy, SEM, ESR, and others. The isotherms of water absorption have been obtained and the deviation from linearity (the Henry low) was analyzed by the simplified model. For PHB-PLA fibers, the loss weight increments as the reaction on hydrolysis are symbate to water absorption capacity. It was shown that the ozonolysis of blend fibrils has a two-stage character which is typical for O3 consumption, namely, the pendant group’s oxidation and the autodegradation of polymer molecules with chain rupturing. The first stage of ozonolysis has a quasi-zero-order reaction. A subsequent second reaction stage comprising the back-bone destruction has a reaction order that differs from the zero order. The fibrous blend PLA/PHB ratio affects the rate of hydrolysis and ozonolysis so that the fibers with prevalent content of PLA display poor resistance to degradation in aqueous and gaseous media.