Yes, it’s entirely possible for a country to run exclusively on renewable energy. Portugal just did it, and it ran for four straight days last week.
According to a recent report: "The consumption of electricity in Portugal was fully covered by renewable sources for more than four days (more precisely, between 6:45 on May 7 and 17:45 May 11).”
Back in the first quarter of 2013, the country was able to successfully run on 70% renewable sources of energy; and in 2014, it was running on 64%.
Portugal’s renewable energy sources mostly come from biofuels and waste. But given Portugal’s ideal weather conditions, wind and hydropower also contribute greatly to the country’s ability to harness renewable energy more efficiently. Of course, location and size also help to make this process easier. For larger, more industrial nations, such changes could not be so easily instituted.
Portugal's milestone comes after Germany announced it was able to source 95% of its energy on renewables last week.