$re = qr{
        (
            (?:
                (?&OPEN)
                (?:
                    (?>(?:.*?(?=(?&OPEN)|(?&CLOSE))))
                    |
                    (?-1)
                )*
                (?&CLOSE)
            )
        )
        (?(DEFINE)
            (?<OPEN>\bBEGIN\b)
            (?<CLOSE>\bEND\b)
        )}x;
[download]

You're my hero!

Now to figure out exactly how it works!

#!/usr/bin/perl

use strict; # https://perlmonks.org/?node_id=11112811
use warnings;

my $input = <<ENDOFSTRING;
beginning a bend
before stuff
begin
  one
  begin
    three
  end
  five
end
after stuff
beginning a bend
ENDOFSTRING

print $input, '-' x 70, "\n";

my $level = 0;
my $nested = $input =~ s/(\bbegin\b)|(\bend\b)|./
  ( $1 && $level++, $level ? $& : '', $2 && $level--)[1] /gesr;

print "$nested\n";
[download]

Outputs:

beginning a bend
before stuff
begin
  one
  begin
    three
  end
  five
end
after stuff
beginning a bend
----------------------------------------------------------------------
begin
  one
  begin
    three
  end
  five
end
[download]