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High-performance cluster-jet targets are ideally suited and applied since years in hadron and laser plasma physics. Therefore, the forthcoming MAGIX experiment at the future energy recovering electron accelerator MESA will use a cluster-jet target to perform high precision measurements on electron scattering experiments, i.e., determination of the proton radius. For this purpose, a cluster-jet target was designed, built up and set successfully into operation at the University of M\"unster considering the requirements of the experimental setup of MAGIX. The details on these requirements, calculations to their realization, e.g., on the nozzle geometry and stagnation conditions of the target gas, their technical implementation and the features of the target which make the target a powerful state-of-the-art target, are highlighted in this publication. Furthermore, the measured and analysed jet beam characteristics from this target using a Mach Zehnder interferometer are presented and discussed. These are of highest interest for the final design of the complete experimental setup of MAGIX. Moreover, first measurements from commissioning beam times performed with the target installed at the already running MAinzer MIkrotron will be presented., Comment: 9 pages, 8 figures, submitted to Nucl. Instr. Meth. A