We report the analysis of microlensing event OGLE-2017-BLG-1038, observed by the Optical Gravitational Lensing Experiment, Korean Microlensing Telescope Network, and Spitzer telescopes. The event is caused by a giant source star in the Galactic Bulge passing over a large resonant binary-lens caustic. The availability of space-based data allows the full set of physical parameters to be calculated. However, there exists an eightfold degeneracy in the parallax measurement. The four best solutions correspond to very-low-mass binaries near (M_1_=170_-50_^+40^M_J_ and M_2_=110_-30_^+20^M_J_), or well below (M_1_=22.5_-0.4_^+0.7^M_J_ and M_2_=13.3_-0.3_^+0.4^M_J_) the boundary between stars and brown dwarfs. A conventional analysis, with scaled uncertainties for Spitzer data, implies a very-low-mass brown-dwarf binary lens at a distance of 2kpc. Compensating for systematic Spitzer errors using a Gaussian process model suggests that a higher mass M-dwarf binary at 6kpc is equally likely. A Bayesian comparison based on a galactic model favors the larger-mass solutions. We demonstrate how this degeneracy can be resolved within the next 10years through infrared adaptive-optics imaging with a 40m class telescope.