How is the height of the Great Red Spot measured? This answer states that the Great Red Spot is getting taller. This means that we are able the determine the spot's height. How do we do this?
The Great red spot looks oval from above but is actually saucer shaped as the storm curves towards the center and the deepest point lies at the center. It is 16350 km wide, so the height is negligible compared to the width. The storm has its roots beneath the surface of the atmosphere. NASA’s Juno spacecraft calculated that the storm penetrate about 300 kilometers into the planet’s atmosphere.
Scientist also calculated the &quot;thickness&quot; of the storm which I believe the portion of storm that is above the atmosphere. The calculated thickness value agreed well with measurements at the cloud level since the Voyager mission in 1979 and was same during the Juno space mission indicating the thickness is more or less constant and is not affected from the apparent shrinking of the spot. The scientists calculated that the Great Red Spot is about 105 miles (170 kilometers) thick.
The cloud cover around the center of the spot vary by about 30 km and at the extreme end, the clouds hovers at a higher position which is about 8 km above the surrounding clouds. The difference is apparent in the false color Galileo image:

The deepest clouds are dark blue, higher clouds are light blue, high thin hazes are pink and high thick clouds are white.
Scientists were able to calculate the dimensions and dynamics of the Jupiter spot by performing a simple experiment. They investigated the dynamics of large Jupiter vortices by performing numerical simulations from a simple laboratory experiment involving a salt-water-filled Plexiglas tank that measured 20 inches by 20 inches by 28 inches (50 by 50 by 70 centimeters). You can find more information in ref. 4.
References:

https://earthsky.org/space/juno-probes-depths-of-jupiters-great-red-spot
https://www.space.com/jupiter-great-red-spot-shrinking-thickness-steady.html
https://www.sciencedaily.com/terms/great_red_spot.htm
Lemasquerier, D., Facchini, G., Favier, B. et al. Remote determination of the shape of Jupiter’s vortices from laboratory experiments. Nat. Phys. 16, 695–700 (2020). https://doi.org/10.1038/s41567-020-0833-9
Space: From Earth to the Edge of the Universe, Carole Stott, Robert Dinwiddie, Giles Sparrow, Dorling Kindersley Ltd, 2010