You can use symbolic references if you switch off "strict refs":

for (1..3) {
    no strict 'refs';
    my $ref = \${'str'.$_}; # this creates a hard reference VIA a symb
+olic reference
    # or:  my $ref = "str$_";  # this is a purely symbolic reference, 
+access is the same as above
    print "The string is $$ref"; 
}
[download]

#!/usr/bin/perl -l

use warnings;
use strict;

our $str1 = qq[A-M o'Foo'];
our $str2 = qq[.,rtyu_''];
our $str3 = qq[$<%^];
our $str4 = qq[this works];

my $ref4 = \$str4;

print $$ref4;  # works as expected

for (1..3) {
    no strict 'refs';
    my $ref = \${'str'.$_};
    print "The string is $$ref"; 
}
[download]

Note that in your example using a simple array would be much easier, less error-prone and more efficient. Symbolic references are almost only useful these days for creating subroutines / packages at runtime (basically, wherever the only alternative is to use eval STRING).

updated again, added some comments