All this article does is point out that if you use that (flawed) method to value Yahoo, and then use the same method to value Yahoo Japan, and then use even less-reliable means to value the non-public Alibaba, and then subtract the last two from the first, you get an unexpected number.
And even if the assumptions underlying the "negative" valuation of the core business were inassailable, its quite easy to have a business to have a negative $10 billion value. Suppose a business has $11 billion in total liabilities, and $1 billion is total assets. Voila, Owner's Equity is -$10 billion.
And this can still be a profitable company. Obviously, making profits means that profit stream has a value (the current value of the stream of future income), but that doesn't mean that you don't have liabilities that exceed that (and the valuation of that stream is not just based on current profit, but expectation of its future continuation. A company can be profitable but the market can lack confidence that it will continue to be profitable.)