• Your code is very strangely indented, quite hard to read. You may want to use perltidy and/or a better editor.
• It actually looks like someone ported a shell script, sprinkled some parts of a C program over it, and copied some pieces of ancient perl scripts of dubious origin. Yes, perl will understand what you want, but it does not look very perlish and is not very efficient.
• Your code uses lots of global variables where a local scope (see my) is sufficient.
• use strict; and use warnings; are missing
• $profile = `\. ~/.profile`;  # Load user profile does not do what the comment says. And it is most likely completely useless.
• eval is useless in $log_for_days = eval(86400 * 91);, in $dcnt=eval(@data-1); and all other places.
• &date_calc; and &get_dirs; is Perl4-style. Avoid the ampersand <UPDATE>when calling functions</UPDATE>, it usually does not do what you intent to do.
• Your code use BAREWORD file handles and the insecure two-argument open, better use the three-argument form and scalar file handles. Also use or instead of || here. And use $! instead of $^E: open my $dirfile,'<','/opt/scripts/dirs') or die "Can't open /opt/scripts/dirs: $!";
• Neither the i flag nor the g flag are needed for matching # in the lines read from /opt/scripts/dirs.
• get_dirs could be implemented using grep to both find unique directory names and to remove comments. Returning right after reading the file's contents automatically closes the file handle IF you use a scalar file handle: return grep { !/\#/ && !$seen{$_}++ } <$dirfile>;
• Using backticks (``) introduces a lot of security problems, especially if you run the script as root. Shells don't always parse data pasted into backticks as you intent. Try to use perl functions instead: `rm -f ...` should be replaced by unlink or by the remove_tree() function from File::Path. `cat ...` should be replaced by open, readline alias <$handle>, close. `ls` should be replaced by opendir, readdir, closedir, or maybe glob.
• @array=`cat ...` reads the entire file into memory. If your log files are large, you may quite easily run out of memory. After that, you iterate over the array line by line. Better use open, as explained above, and use a while loop to read the file line by line. This way, you have only one line in memory, even if your log file contains terabytes of data:

  open my $handle,'<',$filename or die "Can't open filename: $!";
  while (my $line=<$handle>) {
      chomp $line;
      if ($line is interesting ...) {
          # do something with $line ...
      }
  }
  close $handle;
  [download]

• @encdir=`ls .... 2>&1` is even worse than just @encdir=`ls ...`, you end with error messages as directory names in @encdir. You REALLY don't want that, use opendir/readdir/closedir or glob, as explained above.
• You are doing really weird things with arrays and strings, stringifying arrays just to split them back into arrays. Avoid that. Perl functions can return arrays, don't pass function results around by writing to global variables.

Alexander