Jupiter’s dynamics shapes its cloud patterns but remains largely unknown below this natural observational barrier. Unravelling the underlying three-dimensional flows is thus a primary goal for NASA’s ongoing Juno mission, which was launched in 2011. Here, we address the dynamics of large Jovian vortices using laboratory experiments complemented by theoretical and numerical analyses. We determine the generic force balance responsible for their three-dimensional pancake-like shape. From this, we define scaling laws for their horizontal and vertical aspect ratios as a function of the ambient rotation, stratification and zonal wind velocity. For the Great Red Spot in particular, our predicted horizontal dimensions agree well with measurements at the cloud level since the Voyager mission in 1979. We also predict the Great Red Spot’s thickness, which is inaccessible to direct observation. It has remained surprisingly constant despite the observed horizontal shrinking. Our results now await comparison with upcoming Juno observations.