The relevant documentation is Symbolic references in perlref. One important thing to understand is that you cannot interact with lexical variables using symbolic references -- one reason your code won't work is because you can't see your set of specific variables. Correcting your example code,

use strict;
use warnings;

my $var_name_modifier1 = '100';
my $var_name_modifier2 = '200';


our $specific_200_var = "this is from 200\n\n";
our $specific_100_var = "this is from 100\n\n";



print "\n\n\n\n";

print 'this is what we have for 100: '; 
NO_STRICT_BLOCK: {
    no strict "refs";
    print ${"specific_${var_name_modifier1}_var"};
}
print " \n\n";
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Note the (likely) unexpected behavior seen for:

use strict;
use warnings;

our $value = "Valuable information";

print 'this is what we have: '; 
NO_STRICT_BLOCK: {
    my $value = 'Ha ha, I changed it!';
    no strict "refs";
    print ${"value"};
}
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