Gaia16aye was a binary microlensing event, the first such event ever discovered in the direction towards the Northern Galactic Disk and one of the first microlensing events detected and alerted by the Gaia space mission. Its light curve exhibited five distinct brightening episodes, reaching up to 11mag, and was covered in great detail with almost 25000 data points gathered by a network of telescopes. We present the photometric and spectroscopic follow-up covering 500 days of the event evolution and search for a possible microlensing model in order to derive the parameters of the lensing binary system. For Gaia16aye event we employed a full Keplerian binary orbit microlensing model combined with the Earth and Gaia motion around the Sun, to reproduce the complex light curve. The photometric data allowed us to solve the microlensing events entirely and to derive the complete and unique set of orbital parameters of the binary lensing system. We also report on the detection of the first ever microlensing space-parallax between the Earth and Gaia located at L2. The binary system properties were derived from microlensing parameters and we found that the system is composed of two main-sequence stars with masses 0.570.05 M and 0.36+/-0.03M_{sun}_ at 780pc, with an orbital period of 2.88 years and eccentricity of 0.30. We also predict the astrometric microlensing signal for this binary lens as it will be seen by Gaia as well as the radial velocity curve. Events like Gaia16aye indicate the potential for the microlensing method to probe the mass function of dark objects, including black holes, in other directions than the Galactic bulge. This case also emphasises the importance of long-term time-domain coordinated observations which can be done with a network of heterogeneous telescopes.