The Gemini Multi-conjugate adaptive optics System (GeMS) is a facility instrument for the Gemini-South telescope. It delivers uniform, near-diffraction-limited image quality at near-infrared wavelengths over a 2 arcminute field of view. Together with the Gemini South Adaptive Optics Imager (GSAOI), a near-infrared wide field camera, GeMS/GSAOI's combination of high spatial resolution and a large field of view will make it a premier facility for precision astrometry. Potential astrometric science cases cover a broad range of topics including exo-planets, star formation, stellar evolution, star clusters, nearby galaxies, black holes and neutron stars, and the Galactic center. In this paper, we assess the astrometric performance and limitations of GeMS/GSAOI. In particular, we analyze deep, mono-epoch images, multi-epoch data and distortion calibration. We find that for single-epoch, un-dithered data, an astrometric error below 0.2 mas can be achieved for exposure times exceeding one minute, provided enough stars are available to remove high-order distortions. We show however that such performance is not reproducible for multi-epoch observations, and an additional systematic error of ~0.4 mas is evidenced. This systematic multi-epoch error is the dominant error term in the GeMS/GSAOI astrometric error budget, and it is thought to be due to time-variable distortion induced by gravity flexure. ; Comment: 16 pages, 22 figures, accepted for publication in MNRAS