The Earth’s gravitational force does not become zero at any distance. However, it does become very weak at great distances. The gravitational force between two objects is inversely proportional to the square of the distance between their centers of mass. This means that if you double the distance between two objects, the gravitational force between them is four times weaker.
At the surface of the Earth, the gravitational acceleration is about 9.8 m/s^2. This means that if you drop an object, it will accelerate downwards at 9.8 m/s^2. However, if you were to travel far enough away from the Earth, the gravitational acceleration would eventually become so weak that it would be negligible.
At an infinite distance from the Earth, the gravitational acceleration would be zero. However, it is impossible to travel to an infinite distance, so the Earth’s gravitational force will never actually become zero.
In practice, the Earth’s gravitational force becomes weak enough to be negligible at distances of several thousand kilometers. This is why astronauts in orbit around the Earth are weightless. They are still technically being pulled towards the Earth by gravity, but the gravitational force is so weak that it cannot overcome the inertia of their motion.