There's a bit more of an explanation in perldiag for "Out of memory during request for %s" and I think this is presuming that you've compiled using perl's malloc.

    Out of memory during request for %s
        (X)(F) The malloc() function returned 0, indicating there was
        insufficient remaining memory (or virtual memory) to satisfy the
        request.

        The request was judged to be small, so the possibility to trap it
        depends on the way perl was compiled. By default it is not
        trappable. However, if compiled for this, Perl may use the contents
        of $^M as an emergency pool after die()ing with this message. In
        this case the error is trappable *once*, and the error message will
        include the line and file where the failed request happened.

So iff you've got a perl with perl's malloc and you give a buffer to $^M and then run out of memory perl will die and write its error message to that buffer. In that case yes then you could trap that one exception, and you could see where the exception occurred (as opposed to just getting an "Out of memory" error from the OS' malloc maybe).

As far as testing I *think* you could use something like ulimit -m to artificially lower how much memory's available to child processes. Allocate a small buffer as shown in the example, then try and create a large string (run ulimit -m 512 then try perl -E '$^M=(q{}x(1<<16));eval { $x = (q{FOO} x (512 * 1024 * 1024)); }; if($@){die $@;}')

Honestly I don't know how to simulate "running out of memory" for experimenting

That is one of those things that seems like it should be quite easy...that is, until you start to think about actually doing it!
If I were going to try, I think I would start with a virtual machine with very little memory allocated and then get my program to deliberately create lots of arrays of arrays.