my $hash = {
  foo => { one   => [ 1 .. 3 ], },
  bar => { two   => [ 1 .. 6 ], },
  baz => { three => [ 1 .. 9 ], },
};

my @sorted;
foreach my $s (keys %$hash) {
  push @sorted, map [ scalar @{$hash->{$s}{$_}}, $s, $_ ],
                keys %{ $hash->{$s} };
}

print join(" ", @$_), $/
  for sort { $a->[0] <=> $b->[0] } @sorted;

__output__

3 foo one
6 bar two
9 baz three