use strict;
use warnings;

my $emailNum;
my $outFile;

$/ = '';    # Set readline to "Paragraph mode"

while (<DATA>) {
    if (!$emailNum || /^From:/im) {
        close $outFile if $outFile;

        my $fname = sprintf "mails_%06d.txt", ++$emailNum;

        open $outFile, '>', $fname or die "Can't create $fname: $!\n";
    }

    print $outFile $_;
}