The Legend of Korra takes place 70 years after Avatar: The Last Airbender. That time gap gives the world a markedly different political, technological, and cultural landscape when Korra's story begins.
In the intervening decades, Avatar Aang and Fire Lord Zuko helped establish the United Republic of Nations from former Fire Nation colonies. Its capital, Republic City, is a modern metropolis influenced by early twentieth-century industrial design, with cars, radios, electricity, and pro-bending arenas. The old four-nation structure still exists, but international life has become more connected and urban.
The opening story follows Korra, the next Avatar after Aang. She is 17 when she travels to Republic City to study airbending with Tenzin, Aang's son. This succession also helps explain why familiar characters from the earlier series appear as older adults or are remembered through their children.
The 70-year figure describes the setting relative to the events of the original series. It is not the same as every individual character's age or the exact interval from a particular flashback.