There is such a metric - it's called national wealth [1] and it functions much like market cap on a national level. The mechanisms are different as countries can't be "valued" because they can't be bought and as such have no market price. Regardless, economic health is taken into account because the underlying asset values ARE calculated based on market price. National wealth is largely seen as an indicator of ability to pay off debt because it provides a tax base.

As an example, the national wealth of the US in 2014 was $81.8 trillion, while our GDP in 2013 was $15 trillion. So your intuition is correct; GDP is a horrible way to value a country. Though in practice, GDP tends not to vary a lot in mature economies (i.e. inflation tends to stay roughly the same year to year), so you can do a relatively simple NPV calculation on the GDP to approximate the national wealth.

[1] http://en.wikipedia.org/wiki/National_wealth

There is actually. When a country issues bonds you can see some of the future value of the country's output reflected in the interest rate they can get their loans at.