If your records can't be multiline, as your regex seems to imply, you can read the file line by line and populate a hash like this:

use v5.14;
use Data::Dumper;

my %header;
LINE: while (<DATA>) # reading line by line
{
  chomp; # remove the \n at the end
  if (/^(H\d\d)(.*)/) # ^ beginning of string
  {
    $header{$1} = $2;
  }
  else
  {
    last LINE; # Stop running the LINE loop
  }
}

print Dumper \%header;
__DATA__
H00
H01 Hi
H02 Hello
H03 Bonjour
D01
D02
[download]

If your records can be multiline, you may want to read about $/ or the input record separator, and the /m modifier (which will make ^ match the beginning of any line, not just the beginning of the string). The logic may still be the same.