Assuming your filehandle is called $fh, then this is quite nice:

use strict;
use Data::Dumper;

my $count = 0;
while (my @both = grep length, ~~<$fh>, ~~<$fh>)
{
    chomp @both;
    print Dumper \@both;
}
[download]

It might be even nicer if the grep length weren't necessary, but unfortunately it is.