In this paper, we explore the possibility of using natural astrophysical sources to accelerate spacecrafts to relativistic speeds from a conceptual standpoint. We focus on light sails and electric sails, which are reliant on momentum transfer from photons and protons, respectively, because these two classes of spacecrafts are not required to carry fuel on board. The payload is assumed to be stationed near the astrophysical source, and the sail is subsequently unfolded and activated when the source is functional. By considering a number of astrophysical objects such as massive stars, microquasars, supernovae, pulsar wind nebulae, and active galactic nuclei, we show that terminal speeds approaching the speed of light might be realizable under idealized circumstances provided that sufficiently advanced sail materials and control techniques exist. We also investigate the constraints arising from the sail's material properties, the voyage through the ambient source environment, and the passage through the interstellar medium. While all of these considerations pose significant challenges to spacecrafts, our analysis indicates that they are not insurmountable in optimal conditions. Finally, we sketch the implications for carrying out future technosignature searches.