If your perl is of version 5.10 or later, you can add

use 5.010;
[download]

at the start of your script, and substitute all your  print "something\n"; lines with say "something". Looks like a small code, but it does make it more fun to read and write.

As for your original question, you can write something like

    if ($input =~ /^(y|yes)$/) {
        say "Backup is beginning...";
        last;
    }
    elsif ($input =~ /^(n|no)$/) {
        die "Program is shutting down.\n";
    }
    else {
        say "This is not a valid answer.";
        say "Please enter 'yes' or 'no'";
    }
[download]

This uses regexes to test for several values at once, laading to fewer branches with duplicate code. You could also write

    if ($input eq 'y' || $input eq 'yes') {
        say "Backup is beginning...";
        last;
    }
[download]

if you don't want to use regexes.

Finally you should check the return value of system. Just because the backup file exists doesn't mean the backup was successfull. There could be a full file system, file system corruption, permission errors or other problems -- in which case tar will return a non-zero return code, which you should catch. See system for the full documentation on how to do that.