In your C++ (or C) program you have one (or more) allocators. These are just pieces of code that juggle blocks of memory into smaller chunks for the program to use. Typically the allocators get their memory from the OS in pages using some OS system call such as sbrk or mmap.
For the sake of argument, let's say I write an allocator that has a limit of 2MiB, while my system has 64Gib or RAM. The allocator can then fail some request when it's internal 2MiB has been exhausted. In C world it'd return a nullptr. In C++ world it would normally throw bad_alloc.
If this happens does this mean the process is out of memory? Or the system is out of memory? No, it doesn't.
That being said where things get murky is because there are allocators that in the absence of limits will just map more and more pages from the OS. The OS can "overcommit" which is to say it gives out more pages than can actually fit into the available physical memory (after taking into account what the OS itself uses etc). And when the overall system memory demand grows too high it will just kill some arbitrary process. On Linux this is the infamous OOM killer that uses the "niceness" score to determine what to kill.
And yes, for the OOM killer there's very little you can do.
But an allocation failure (nullptr or bad_alloc) does not mean OOM condition is happening in the system.