Which binary star system has a greatest angular distance? Visible from the Earth, which binary star system has greatest angular distance ?
Wide binary stars it turns out is an active research topic, with groups using statistical searches of catalogs (Tycho Gaia ...) to find in 3D space pairs of stars that are moving together taking into account possible orbital motion. Here people are looking at the physical separation not the angular separation. On the theory side it is suggested that nearly all stars were formed as part of a binary. So somewhere out there is sister of our Sun. One paper looked at nearby stars. They find several very wide binary pairs visible to the eye plus more fainter. In particular they mention Capella and 50 Peg which have an angular separation of 20 deg. This they class as an escaped binary : so no longer in orbit (thanks Connor for pointing this out).
There is however Fomalhaut; a faint (mag 12.7) 3rd member of the system was discovered in 2013. It is 5.7 deg away from the primary. The other companion is  2deg away in another direction. The whole system spans 8 deg. The Sky and Telescope article claims this as the widest (on sky) binary, beating Alpha AB Proxima system.