Moving the Earth is dealt with in some (err, much) depth here: http://qntm.org/moving

In terms of Archimedes:

"Give me a place to stand, and I will move the Earth", proclaimed Archimedes upon discovering the principles of leverage. Unfortunately, that place needs to be immovable, which is impossible. Also, what he didn't specify is that he also needs an equally immovable fulcrum - equally impossible - and a lever of stunning length which is unbreakable. Hah! Nice try, Archimedes.

"Unfortunately, that place needs to be immovable"

No, it merely needs to be of sufficient mass. As a practical strategy for moving the earth, this would clearly be infeasible, but, strictly speaking, I don't think that it is impossible.

When Newton came along we realized that all we needed to do was jump. Granted, we don't move the earth very much, and it comes right back, but still...