I remember a short story, which initially appears to be set on Earth, wherein a Space Shuttle is sent up to collect an alien probe that’s entered orbit. It eventually reveals that it came from Earth, and that the last launches of these probes were automatic - its creators were all dead.
BTW if Wiki’s timeline of the far future is correct, Earth will only be able to support life for less than a billion years.
Not really. Imperial Earth is only set 300 years in the future (during the Quincentennial of the USA in fact), and there’s a thriving, active civilization in the Solar System. Sure, they haven’t left the Solar System (yet), but there are literally hundreds or thousands of sci fi stories with that setting. I think the OP is looking for something more specific.