Are those 1s and 0s bits or are they just characters? If the latter, you could just read one byte at a time from each file, then write them out:

    open FH1, "what.txt" or die $!;
    open FH2, "what1.txt" or die $!;

    my($buf1, $buf2, $total);
    while (read(FH1, $buf1, 1) && read(FH2, $buf2, 1)) {
        $total .= $buf1 . $buf2;
    }

    close FH1;
    close FH2;
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Of course, you'd get faster results by reading in 1000 bytes or so at a time, then grabbing the first byte of each buffer. That would cut down on disk operations.