There are differences, firstly, between still and stable. If you want a video to be still, as if it were filmed on a tripod, then you can use a tracking stabilisation technique. This is when you track a point in the background, and since the background is meant to remain motionless when on a tripod, the computer can compensate by moving the video in the opposite direction to the motion. By zooming in, you can remove the bars at the side that would appear from moving the image out of the screen.
However, if you want steadiness, where the camera still moves, but without violent shaking, such as if you were using a Steadicam, and have the privilege of using professional software such as Premiere, there are methods to achieve this. In Premiere, you can use the 'Warp Stabiliser' effect, and if you ensure that the 'effect' drop-down is set to Smooth Motion instead of No Motion, you should be good to go. But remember that if there is too much shake in the camerawork, Warp Stabiliser could make it look even worse. If you do not have access to Premiere, you can use the previous technique of removing motion completely and adding something like Cinecom's handheld camera effect package in Hitfilm to simulate handheld motion, smoothly.