How is it possible for an object to fall towards Earth, but to get no closer to Earth's surface?

In the first answer, SammyVazzy is on the right track, but then

he gang aglay, straight into left fayld.

Orbiting has nothing to do with the Earth moving away. It's that the

Earth is a sphere ... shaped like a ball. So, in the time the satellite

falls a few feet, it also moves sideways a few feet, and the Earth's

surface under it also curves away from it, so it doesn't lose any altitude.

The satellite is constantly falling, but the Earth's shape is constantly

falling away from it.

IF the satellite were not moving sideways, then it would sink into the

atmosphere like a rock, just as you'd expect, and either burn up or go

kerplunk onto the land or into the ocean.

This would be called being in Obit, if you are in Orbit of Earth, then you are essentially Falling twords earth but because earth Moves and is not stationary, you Miss the Earth as you fall. So if your in Orbit of something, you are falling very fast but the object you are orbiting is moving away from you even faster.