The general idea was that we can orbit a fixed reference (In this case the earth) at a high relative speed. At high earth orbit (35km), orbiting the circumference (Simplifying assuming a perfect circle) would mean one orbit done per second would be 2/3c.

The time delta math (Sqrt(1-v^2/c^2)) puts us at 3 earth days per two space days.

The vacationing was more cheeky, in that in this case a 2 week vacation would appear 3 weeks on earth. If there would be some way to quickly switch between modes than 1 week, you would spend more self-referential time doing something than your home clock would count.