This was a more elegant solution before I started testing it. This uses the flip flop operator ("..", see perlop) to detect when it's in the delimiters.

my $start = qr/A{6}/;
my $end   = qr/B{6}/;
while ( <DATA> ) {
    if ( m{$start} .. m{$end} ) {
        next if m{$start};
        next if m{$end};
        print;
    }
}
__DATA__
hello world0
hello world1
hello world2
AAAAAA
this is a test of the
emergency broadcast system
BBBBBB
hello world0
hello world1
hello world2
AAAAAA
if this had been a real
emergency more instructions
would have followed
BBBBBB
hello world0
hello world1
hello world2
__END__
this is a test of the
emergency broadcast system
if this had been a real
emergency more instructions
would have followed
[download]

This doesn't produce the blank lines between the text sections like the original one does, but it pulls out the desired text clearly enough. If the blank lines are important, you could do that near one of the nexts.