  Let G be a group acting on the plane by orientation-preserving
homeomorphisms. We show that if for some k>0 there is a ball of radius r >
k/\sqrt{3} such that each point x in the ball satisfies |gx -hx| < k for all g,
h in G, and the action of G satisfies a nonwandering hypothesis, then the
action has a global fixed point. In particular, any group of measure-preserving
orientation preserving homeomorphisms of the plane with uniformly bounded
orbits has a global fixed point. The constant k/\sqrt{3} is sharp. We also show
that a group acting on the plane with orbits bounded as above is left
orderable.
