Orbit around game object

  public Transform target;

    void Update()
    {
        Vector3 relativePos = target.position - transform.position;
        Quaternion rotation = Quaternion.LookRotation(relativePos);

        Quaternion current = transform.localRotation;

        transform.localRotation = Quaternion.Slerp(current, rotation, Time.deltaTime);
        transform.Translate(0, 0, 3 * Time.deltaTime);
    } 

First the givens.

relativePos is the vector direction from to target. LookRotation is a function that derives a Quaternion from a vector you'd like your object to face towards. Slerp Spherically interpolates rotation between two rotations meaning it rotates from a given rotation to another smoothly.

Now to the explanation.

It revolves around the target because of the transform.Translate without it your object will just rotate to face the target (because of the LookRotation) on its own axis with no movement. In the tranform.Translate the third parameter (3 * Time.deltaTime) means move the object forward along its z axis 3 units/second therefore it revoles because it's constantly trying to move 3 units/second on the z axis but the Slerp keeps pulling it in making it rotate towards the target so tranform.Translate moves it and Slerp keeps rotating it back to target which results in orbiting.

And you can't just use transform.RotateAround() because

1. You won't be able to configure which way your object faces it'll just revolve around a target. You can use it in combination with transform.LookAt() but that'll result in some jittery effects while on the code above Slerp Spherically interpolates rotation which basically means smoother rotation and less jittery.
2. You won't be able to specify rotation radius with transform.Translate

The object runs away because in tranform.Translate the third parameter (3 * Time.deltaTime) means move the object forward along its z axis 3 units/second while -3 means move it backward therefore it runs away But if you look closer it's still facing the target. Instead of moving back on the Z axis it moves back along the direction facing the target because of the LookRotation and Slerp functions.

I hope this explained it well if you have any more questions/need more clafication just reply and I'll get back to you.