Is it not possible to make a C++ application "Crash Proof"?

You can check the bounds of an array in C++, std::vector::at does this automatically.

This doesn't make your app crash proof, you are still allowed to deliberately shoot yourself in the foot but nothing in C++ forces you to pull the trigger.

No. Even assuming your code is bug free. For one, I have looked at many a crash reports automatically submitted and I can assure you that the quality of the hardware out there is much bellow what most developers expect. Bit flips are all too common on commodity machines and cause random AVs. And, even if you are prepared to handle access violations, there are certain exceptions that the OS has no choice but to terminate the process, for example failure to commit a stack guard page.

By crash I primarily mean forceful termination by the OS upon memory access violation, due to invalid input passed by the user (like an abnormally short junk data).

This is what usually happens. If you access some invalid memory usually OS aborts your program.

However the question what is invalid memory... You may freely fill with garbage all the memory in heap and stack and this is valid from OS point of view, it would not be valid from your point of view as you created garbage.

Basically - you need to check the input data carefully and relay on this. No OS would do this for you.

If you check your input data carefully you would likely to manage the data ok.