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As far as I remember there was "уголок", not "закуток" in the original Vysоtsky's song.
"(самый дальний) уголок" (here) - means "a place far, far away, a God's forgotten hole."
But "закуток" means...

"ростовчане" - inhabitants of Rostov (Rostov-on-Don) city. But in this case, "ростовчане" are the passengers of the flight to Rostov.

"позарез" - very urgent (slang)

"закуток" - in this case,...