Is it posible get distance of object from picture with static camera?
Yes, but it's gonna be difficult. You need the intrinsic camera matrix and a depth map. If you don't have them, then forget it. If you only posses a RGB image then it won't work.
How would you find the height of objects given an image?
Yes it is possible. What you are describing has an entire industry around it, called Photogrammetry
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