I think you’re confused about where the force are applied. When you push against an object, you exert a force on the object, but the object push a reaction force against YOU, not the object. So both move. From the point of view of a stationary outsider, the entire you-object can be considered a single system, in which case the total force on the system is indeed 0, and in fact the system doesn’t move (its center of mass is stationary), it merely stretches out.
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