Poly(glycolic acid) (PGA) is a rapidly degradable polymer mainly used in medical applications, attributed to its relatively high cost. Reducing its price will boost its utilization in a wider range of application fields, such as gas barriers and shale gas extraction. This article presents a strategy that utilizes recycled PGA as a raw material alongside typical carbon nanomaterials, such as graphene oxide nanosheets (GO) and carbon nanotubes (CNTs), to produce low-cost, fully degradable yarns via electrospinning and twisting techniques. The results demonstrate that the tensile strength of the PGA/GO composite yarn increased to 21.36 MPa, and the elastic modulus attained a value of 259.51 MPa with a 3 wt% of GO loading. The addition of an appropriate amount of GO enhances the tensile resistance of the composite yarns to a certain extent. However, excessive application of GO and CNTs can lead to surface defects in the nanofibers, reducing their mechanical properties. Moreover, the integration of both materials could inhibit the degradation process of PGA to some extent, thereby partially addressing the issue of excessive degradation rates associated with the relatively low molecular weight of recycled PGA.