Here is a more "perl-ish" way to do this, using a perl "array" (@seq).

"push" adds stuff into the back of an array, while "pop" removes it.
"chomp" deletes the last "line ending" character(s), if they exist.
The "sort" uses the "Numeric compare" operator "<=>" to compare the length, and returns each item in the sorted array as "$_", which is then printed along with a newline (\n).
The sort BLOCK using "$a <=> $b" is a common/canonical perl idiom for numeric sort. Just accept that till you research it deeper. "$_" is commonly used as a "temporary" variable when the programmer is too lazy to give it a name.

use strict;
use warnings;

my @seq; # DNA sequence strings

do{
    print "Please enter DNA sequence (END to finish):";
    push @seq , $_=<STDIN>;  # First, assign whatever is read into $_,
+ then append it to @seq
  } until $_ eq "END\n";

pop @seq;   # Remove the extra "END\n" entry;
chomp @seq; # Remove line endings on each entry

print "$_\n" 
  for  sort {length $a <=> length $b} @seq;
[download]

Note - this code is not limited to 4 entries - it requires an END to be entered after the last entry.