HA! I figured it out!

This works, and no filehandles in the error messages:

my @Table;
sub LoadTable
{
    local( $/, $. );
    my $position = tell( DATA );
    @Table = map {chomp; hex} split ' ', <DATA>;   
    seek DATA, $position, 0; # Reset the filehandle, rather then closi
+ng it. 
                             # Never close DATA.
}
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But if I take $. out of the local line, then the filehandle shows up in the error messages. Thanks all!