Has anyone driven under, or received camera footage from, the Great Artesian Basin in Australia?
Groundwater is stored in the porosity of rocks, that is space between clasts or cracks. Larger cavities, e.g. karsts, is usually only a minor contribution to ground water reservoirs and are only common in limestone. 

The Great Artesian Basin aquifer is made up of Mesozoic sandstone. It is covered by impermeable rocks that allow pressure to build up within the aquifer so that as rainwater enters at high elevation it will keep the hydrostatic pressure, even when the topography is lower. 

The only way to see the rocks at great depths is by drilling and, if the budget allows, also keep core sample to be able to analyze it. Important information also comes from borehole logging when geophysical instruments are lowered into the borehole to measure e.g. porosity and density of the rocks. One can also see the continuation of the formation on the surface and study the rocks to understand how they look at great depths. 

This informative document explains some of the features of The Great Artesian Basin rather well, but don't be mislead to think about the aquifers as caves, it is only water soaked rocks.