Another option, instead of closing and re-opening B every time, would be to open it once at the start and use seek to reset the file's position:

open my $fhA, '<', '1.dat' or die $!;
open my $fhB, '<', '2.dat' or die $!;
while (my $lineA=<$fhA>){
    seek $fhB, 0, 0;
    while(my $lineB=<$fhB>){
        print $lineA,' ', $lineB;
    }
}
[download]

Better still, though, would be to slurp B into an array instead of re-reading the file for every line in A:

open $fhA, '<', '1.dat' or die $!;
open $fhB, '<', '2.dat' or die $!;
@B_lines = <$fhB>;
while (my $lineA=<$fhA>){
    for my $lineB (@B_lines) {
        print $lineA,' ', $lineB;
    }
}
[download]

(And I also switched both examples to use lexical filehandles and three-arg open instead of doing it The Old-Style Way because they're better/safer/etc.)