This post from Bustle.com gives some good reasoning for Interstellar being set around the 2070s. In summary: the descriptions of halcyon days gone by from Cooper's live-in father-in-law Donald sound very much like our present day.

Where does Interstellar take place?

Set in a dystopian future where humanity is struggling to survive, the film follows a group of astronauts who travel through a wormhole near Saturn in search of a new home for humanity.

How did 23 years pass in Interstellar?

It's orbiting Gargantua, the massive glowing black hole that exists in the foreign galaxy. Due to Gargantua's massive gravitational pull, “every hour on that planet is seven years on Earth”. After a massive tidal wave hits the spacecraft and delays their exit, they find that 23 years have passed on Earth.