In short, Putin's wrong, or at least oversimplifying drastically. Airport transit areas are exempted from immigration regulations, but they are very much the country's property, under its authority and jurisdiction. As a simple example, if you're transiting via an airport and are caught carrying contraband there, you'll be punished under the transit country's legislation, as a long list of people busted in Changi Airport, Singapore have found out.
All that said, Snowden's definitely a legal edge case. For example, you can only deport someone who has entered the country (legally or illegally), which Snowden has not. Russia could easily extradite Snowden if they wanted to, since Snowden is now subject to Russian jurisdiction and espionage is a crime in both Russia and the US, but Russia and the US do not have an extradition agreement and, more to the point, Russia has no interest in doing so. Russia could always kick him out under some legal pretense (loitering? vagrancy?), but they'd have to find somebody to take him and issue him travel documents first!
As an ordinary traveller, though, I wouldn't lose too much sleep over this. It's very unusual to end up in this position, since the vast majority of people not accepted by their final destination are either refused boarding by the airline at the origin, or deported back to where they came from once they have tried to enter the country. It takes an extraordinary run of bad luck, like being exiled from your country, losing your passport mid-trip and being unable to obtain a new one, to end up a refugee in the transit area.