Once you hit level 8 in the Astronaut career, your daily task becomes "Space Mission"
It took me a while to figure out how to complete it. You have to build a rocket ship. You need a lot of space to do it too, the object itself is pretty large. It also takes a long time to build, so I suggest getting started on it as early as you can afford it.
Once it's completed, you just have to "Explore Space" to complete your daily task.