use strict;
use warnings;
use feature qw(say);
use utf8; # <-- needed, since you have high characters in your source

my $file = q(weird.txt);

open( my $fh, '>', $file );
binmode $fh, ':encoding(UTF-8)';
say $fh qq(nase\ngöre);
close $fh;

say qx (file -I $file);
say qx(echo \$LANG);
say qx(cat $file);

open( $fh, '<', $file );
binmode $fh, ':encoding(UTF-8)';
binmode STDOUT, ':encoding(UTF-8)'; # <-- here
say <$fh>;
close $fh;

__END__