use strict;
use warnings;

use Text::Balanced qw( gen_delimited_pat extract_multiple );

my $text = q{jfjsa as,.n d"fdsafjl"jop'fdsjklf fds'457"fjdsklaoir"jkl4
+5;fs ier987543" fsdjkal"};

my @extracted = extract_multiple($text, [
    gen_delimited_pat(q{'"}),
]);

print "$_\n" for @extracted;
[download]

jfjsa as,.n d
"fdsafjl"
jop
'fdsjklf fds'
457
"fjdsklaoir"
jkl45;fs ier987543
" fsdjkal"
[download]

Text::Balanced

Update: The difference between mine and OGB's is that mine allows escaped delimiters such as 'foo\'s room' and mine doesn't strip the delimiters as per your requested output.