Handling a stack overflow is not the right solution, instead, you must ensure that your program does not overflow the stack.
Do not allocate large variables on the stack (where what is “large” depends on the program). Ensure that any recursive algorithm terminates after a known maximum depth. If a recursive algorithm may recurse an unknown number of times or a large number of times, either manage the recursion yourself (by maintaining your own dynamically allocated stack) or transform the recursive algorithm into an equivalent iterative algorithm
A program that must be “really robust” will not use third-party or external libraries that “eat a lot of stack.”
Note that some platforms do notify a program when a stack overflow occurs and allow the program to handle the error. On Windows, for example, an exception is thrown. This exception is not a C++ exception, though, it is an asynchronous exception. Whereas a C++ exception can only be thrown by a throw statement, an asynchronous exception may be thrown at any time during the execution of a program. This is expected, though, because a stack overflow can occur at any time: any function call or stack allocation may overflow the stack.
The problem is that a stack overflow may cause an asynchronous exception to be thrown even from code that is not expected to throw any exceptions (e.g., from functions marked noexcept or throw() in C++). So, even if you do handle this exception somehow, you have no way of knowing that your program is in a safe state. Therefore, the best way to handle an asynchronous exception is not to handle it at all(*). If one is thrown, it means the program contains a bug.
Other platforms may have similar methods for “handling” a stack overflow error, but any such methods are likely to suffer from the same problem: code that is expected not to cause an error may cause an error.
(*) There are a few very rare exceptions.