How can an object accelerate towards the center without ever getting any closer to the center?

If the object is making circular motion, it can accelerate towards the center while it is moving around the center.

Explanation:

In circular motion, the object is moving on a circular orbit around the center with velocity v. Then the radial acceleration is

This is called the centripetal acceleration.

For example, if you tie a ball with a string and rotate the ball circularly you can feel the tension in the string. This tension is equal to according to Newton's Second Law.