The short answer: No.
The barycenter is the center of mass of two or more bodies that are orbiting each other, which is the point around which they both orbit.
In cases where one of the two objects is considerably more massive than the other (and relatively close), the barycenter will typically be located within the more massive object. Rather than appearing to orbit a common center of mass with the smaller body, the larger will simply be seen to wobble slightly. This is the case for the Earth–Moon system, where the barycenter is located on average 4,671 km (2,902 mi) from the Earth's center, well within the planet's radius of 6,378 km (3,963 mi).
So here we are, on a planet rotating at a thousand miles per hour, and the planet is looping in a 585,000,000 mile orbit around the sun, and our solar system orbits the Milky Way at about 500,000 miles per hour (one complete orbit taking about 226 million years), and the Milky Way is moving through space (while undulating), and space is expanding. That means our path through space—like our path through life—is a wobbly, wiggly spiral within other spirals. It means we’re never in the same place twice.