http://archive.adaic.com/standards/83lrm/html/lrm-11-01.html
"STORAGE_ERROR This exception is raised in any of the following situations: (...) or during the execution of a subprogram call, if storage is not sufficient."
It's not safe though because throwing an exception, panicking, etc, is still a denial of service. It's just more deterministic than silently overwriting the heap instead. If the program is critical then you need to be able to statically prove the full size of the stack, which you can do with C and C++ with the right tools and restrictions.
The Ada language specification says the Ada programmer can expect any Ada compiler when used in fully compliant mode to properly raise STORAGE_ERROR when a stack overflow occurs.
Only the Ada compiler writer has to deal with this, not every single programmer on every single program and platform (the UB behaviour of some languages).
In the case of GCC/GNAT the compiler manual provides insight on how to be in compliant mode per target regarding stack overflow, what are the limitations if any. You have tools to monitor and analyze you Ada code in this respect too.
If a segfault leads to some other state you do not deem "safe", such as a single program gating access to a valuable asset with a default fail state of "allow", you just have a fundamental design flaw in your system. The safety problem is you or your AI agent, not the segfault.
> If the program is critical then you need to be able to statically prove the full size of the stack, which you can do with C and C++ with the right tools and restrictions.
Only for specific implementations; and neither C nor C++ standard guarantee that your program won't overflow the stack anyway. (Have a look, the standard says nothing about stack overflows, they still occur in practice with supposedly compliant compilers. So it seems that there are no rules about when they can happen.)