A proof-of-principle experimental setup for the extraction of 6 GeV electrons from the DESY II Booster Synchrotron using the channeling effect in a bent crystal is elaborated. Various aspects of the experimental setup were investigated in detail, such as the particle beam dynamics during the extraction process, the manufacturing and characterization of bent crystals, and the detection of the extracted beam. In order to optimize the crystal geometry, the overall process of beam extraction was simulated, taking into account the influence of radiation energy losses. As result it is concluded that the multi-turn electron beam extraction efficiency can reach up to 16%. In principle this crystal-based beam extraction technique can be applied at any electron synchrotron in order to provide multi-GeV electron beams in a parasitic mode. This technique will allow to supply fixed-target experiments by intense high-quality monoenergetic electron beams. Furthermore, electron/positron crystal-based extraction from future lepton colliders may provide an access to unique experimental conditions for ultra-high energy fixed-target experiments including searches for new physics beyond the Standard Model.