Well I don't think it could be stabilized, but imagine two
equally massive planets initially at rest and separated by
a distance R. A Superball is fired from planet A to planet
B with velocity V* and sufficient momentum to offset the
force of gravity for a time 2R/V. This ball then bounces
elastically back and forth between the two planets,
keeping them separated by R. The station keeping could be
maintained by active intervention. Station Ken Griffey Jr.
on planet A and another player to be named later on planet
B to tune the ball's momentum.