Can a country survive and prosper without international trade or investment from overseas? To what extent can a country be 100% independent from the rest of the world? Is there any country like that in the world? Maybe North Korea, but North Korea receives supports from China.
Besides allowing countries to take advantage of comparative advantage, as mentioned, an open economy can engage in intertemporal trade. 

This possibility is useful because it allows a country to smooth consumption over time, even when domestic output fluctuates. For instance, in good times the economy can save (consume less than actual output, lending to foreign countries), allowing the economy to consume more than actual output in bad times. 
This in fact is welfare improving relative to autarky (in which consumption in any period must equal current output (to simplify assume there is no investment), when we assume that consumer's preferences are convex.

Therefore, I don't know if a country can survive/prosper being a closed economy, but certainly there are many advantages for an economy that engages in international trade.