Why not just change the input string with s/""/"/g and the like. Then you can extract text between single quotes. Below you can see my attempt to extract the quoted data (tested).

#!/usr/bin/perl
use strict;
use warnings;

while (<DATA>)
        {
        chomp;

        # remove the pesky quotes at the beginning and end
        s/^"//;
        s/"$//;

        # change all doubled quotes into just singles
        s/""/"/g;

        print $_ . "\n";
        }
__DATA__
""
"a"""""
"a""b""c""d""f"
"This is a ""test"""
"Here we have """"a nested set of double-quoted quotes"" or whatever."
+""
"Harvey ""the Screwdriver"" Ledbetter"
[download]

Output:

a""
a"b"c"d"f"
This is a "test"
Here we have ""a nested set of double-quoted quotes" or whatever."
Harvey "the Screwdriver" Ledbetter
[download]