Yes, huhuh is right. Earth itself isn't getting stretched out as part of the expansion, and neither is the solar system. That's because they are gravitationally bound, i.e. different parts of Earth/the solar system are pulling on each other via gravity which keeps them from getting carried along with the expansion. Even galaxies mostly aren't getting pulled apart. It is on even larger distance scales that expansion comes into play.
Also I didn't understand this part: "I'm really curious if our solar system is in a static place, or because of lack of gravity it's all in a free fall." I'll just point out that in physics, free fall means moving freely under the influence of gravity. So if you jump off a cliff, there sure is gravity, but (ignoring air resistance) you are in free fall. Likewise, our solar system as a whole is indeed affected by gravity, and is in free fall toward the center of our galaxy (but also with lateral velocity, so it is orbiting). Not sure if that helps answer the question.