Question

An object has an acceleration of 0 m/s². Does this mean that the object must be stationary?

Answer 1

Zero acceleration means that the object's motion is not changing. That tells
you that the object's velocity is constant ... it's moving in a straight line at a
steady speed (which may be zero).

Answer 2

No. Having '0' acceleration means not speeding up in any directions. If an object had speed before it's acceleration became '0' and if the object is not stopped the object will keep moving with constant velocity.