I grew up using the Amiga, when having memory allocation fail was routine (a standard Amiga 500 for example, came with 512KB RAM, and was rarely expanded to more than 1MB, so you
would run out of memory).
What you do when malloc() fails depends entirely on your application: If a desktop application on the Amiga would shutdown just because a memory allocation failed, nobody would use it. The expection was you'd gracefully clean up, and fail whatever operation needed the memory, and if possible inform the user to let him/her free up memory before trying again.
This expectations in "modern" OS's that malloc never fails unless the world is falling really annoys me - it for example leads to systems where we use too much swap to the point where systems often slow down or become hopelessly unresponsive in cases where the proper response would have been to inform the user - the user experience is horrendous: Swap is/was a kludge to handle high memory prices; having the option is great, but most of the time when I have systems that dip into swap, it indicates a problem I'd want to be informed about.
But on modern systems, most software handles it so badly that turning swap off is often not even a viable choice.
Of course there are plenty of situations where the above isn't the proper response, e.g. where you can't ask the user. But even for many servers, the proper response would not be fall over and die if you can reasonably dial back your resource usage and fail in more graceful ways.
E.g. an app server does a better job if it at least provides the option to dynamically scale back the number of connections it handles rather than failing to provide service at all - degrading service or slowing down is often vastly better than having a service fail entirely.