Is it possible to achieve the goal of not having a goal?
Not sure if this is a rhetorical question, but achieving the “goal of not having a goal” is a goal in and of itself. Whenever you desire any outcome, it is technically a goal.
If you meant more like not setting written goals and mapping them out with specific in order to get what you want, then the answer is yes, people do this every day.
But, if you want to be productive, waste less time and hit the target with precision, writing a goal down, being specific about what you want to achieve and mapping your path to achieving the goal helps.