What have you tried so far? Have you looked at the gzip or bzip2 programs? You can compress your file with them and then read from your file by opening a pipe to them:

my $packer = 'bzip2';
my $file = 'data.txt.bz2';

open my $fh, "$packer -cd $file |"
    or die "Couldn't decompress '$file': $!/$?";
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Alternatively, you could encode each of the four characters into two bits, thus storing four characters per byte. I guess this approach won't be more efficient space-wise than the gzip or bzip2 approach, but it retains the ability to do random reading in your file:

use strict;
my %charmap = (
    A => '00',
    C => '01',
    G => '10',
    T => '11',
);

my $string = 'GATTACA';
$string =~ s/(.)/$charmap{$1}/ge;

print "$string\n";

my $compressed = pack 'b*', $string;

print "$compressed\n";
printf "%d bytes\n", length $compressed;

# now use vec() to get at the single parts of $compressed

my $decompressed = unpack 'b*', $compressed;

print "$decompressed\n";
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But have you looked at BioPerl? I'm pretty sure that they have support for that stuff.

Corion's suggestion to use a compression program is a good suggestion. If the sequence represented by the strings are from coding regions, it is likely that some sub-sequences (codons) occur with much higher frequency than other sub-sequences (e.g., stop codons). In this case, certain types of compression algorithms can potentially achieve better compression than the 4:1 you'd get with bit packing.