What makes a state?

Taiwan: flag = country?
I have written in the past about geopolitics, in particular the nation-state quandary: what differentiates a state from a nation, and what do each definition encompass. This time, an article in The Economist in April questions what makes a state a state.
Is an independent state one that is recognised by nations? Is participation in global organisations an indication of statehood? Is there a minimum size (people, land, services) to it before it can qualify as an independent state? To illustrate, here are some comparisons of supposed states that are not universally recognised as states, or may not exactly be construed as states under different definitions.
Kosovo. It is recognised by 65 states as independent of Serbia since 2008, including USA and much of Western Europe, but not by Russia and much of Asia. The European Union and United Nations have not recognised the independence of Kosovo yet, though some countries in the EU and UN have recognised Kosovo, so is it considered a state?
Taiwan. It is recognised by only 23 countries, far cry from its heyday when communist People's Republic of China was shunned in favour of democractic Republic of China, based in Taiwan. It is not represented in the United Nations, but has observer status in many global bodies and is recognised to have a government and some form of de facto "sovereignty". Taiwan is rare amongst disputed "states" for having its own domain name (.tw) and phone prefix (+886).
Vatican City. The smallest state on Earth, at 44 hectares, located inside Rome. However the diplomats of Vatican City work for the pope (Holy See) and not the state itself, and the Holy See is represented in the UN and not Vatican City.
The article in The Economist gives more illustrations and possible definitions that complicate matters, but really, it seems like when asks "What makes a nation?" one should also ask "What makes a state?". Because conventional definitions of statehood with regard to fixed, internationally recognised borders is one that deserves some questioning.