Why can't gravity repel things? Gravity is the result of the curvature of space, so could the curvature actually send objects away from the source?
Apart from the field-theoretical standpoint presented by Stan, one can repel objects in a sense, when taking orbital mechanics into account.

The slingshot maneuver extracts angular momentum and energy of an orbiting mass by the use of gravity. The trick here is that the probe's velocity just gets redirected in the planet's reference frame. But this results in an acceleration in the solar rest-frame that the probe essentially tries to leave.

As you were asking about curvature, I don't know how and if centrifugal potentials that play a role here are described by general relativity.