well, if you always have the > at the beginning, you could do something like that:

my $dna;
{
  local $/ = "\n>";
  $dna = <DATA>;
  $dna =~ s/^>//;
  while (<DATA>) {
    chomp;
    $dna .= $_;
  }
}
chomp $dna;
[download]

update: ok, i forgot you don't want the line starting with > at all. then the approach with reading every second line might be the best, depending on your specification. my example shows you, however, how you can use $/