The project mentioned in the article seems particularly safe to me as they seem to run it with constant pressure, defined by the hydrostatic head of their deeply dug cavities. They don't really have a pressure vessel, they have a flooded cavern where they create a bubble by pushing in air. If you pop off a seal from the pipes connecting compressors/turbines to the bubble could still hurt yourself a lot shooting parts through the room, but that should be all.
The cavity they build is 2000 feet deep, which equates to the pressure of a CO2 cartridge that you'd use for e.g. carbonating a beverage. Not coincidentally at all I think, they either aim for the deepest cavity were hydrostatic pressure isn't enough to liquefy parts of the air, or for the least deep dig that gives them liquefication, where the air "boils off" as they open the valves.
It's not perfect, of course. Accidents still happen, for example: https://en.wikipedia.org/wiki/Taum_Sauk_Hydroelectric_Power_... although fortunately nobody was killed.
https://en.wikipedia.org/wiki/Lake_Peigneur
So I'm picturing water leaking into a salt mine full of pressurized air. Naively it makes a hole, and a hole is just a leak. But if the water starts dissolving the salt in a process that runs away, then it is more like popping a balloon, but on a grand scale.