The growing "Hubble tension" has prompted the need for precise measurements of cosmological distances. This paper demonstrates a purely geometric approach for determining the distances to extragalactic binaries through a joint analysis of spectroastrometry (SA), radial velocity (RV), and light-curve (LC) observations. A parameterized model for the binary system is outlined, and simulated SA, RV, and LC data are computed to infer the probability distribution of model parameters based on the mock data. The impacts of data quality and binary parameters on the distance uncertainties are comprehensively analyzed, showcasing the method's potential for high-precision distance measurements. For a typical eclipsing binary in the Large Magellanic Cloud, the distance uncertainty is approximately 6% under reasonable observational conditions. Within a specific range of data quality and input parameters, the distance measurement precision of individual binary star systems is generally better than 10%. As a geometric method based on the simplest dynamics, it is independent of empirical calibration, and the systematics caused by model selections can be tested using nearby binaries with known distances. By measuring multiple binary star systems or monitoring one binary system repeatedly, geometric distance measurements of nearby galaxies can be achieved, providing valuable insights into the Hubble tension and advancing our understanding of the Universe's structure and evolution.