This could be one way to do it with less code.

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

my $DNA = "AACCGTTAATGGGCATCGATGCTATGCGAGCT";
my @rand = split("", $DNA);

my $num  = @rand;

my $max = 5;
my @word;

for (my $i=0; $i<$max; $i++) {
    my $x = int(rand($num))-1;
    push @word, $rand[$x];
}

print @word;
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By setting the variable $max to the desired length of the out put string.<br/
Of course if you wanted to use the random word as a variable substitute the "print @word" with

my $ranWord = join("",@word);
print $ranWord;
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or something along those lines to create a variable from the array contents