I did check some answers on similiar questions here, but I wasn't satisfied with the answer. When we are sitting (or if there is any body) on merry - go - round there should be some net force which acts radially in to keep us rotating relatively to ground. In this case it is said that static friction is this force since we are not moving with respect to merry - go - around. We are looking at things from inertial reference frame (no inertial forces). As far as I know static friction opposes motion when some outer force tries to move it from relatively to some other surface (to move us relatively to merry - go - round). So, there should be some force acting radially out to cause static friction acting radially in, but even in that case sum of forces acting in radial direction is zero since static friction balances outer force trying to move us. If so, there is no net force acting radially in to allow rotation. Where have I gone wrong?
Friction is the centripetal force in this scenario and aside from gravity it is the only force acting on the person on the Merry-go-round.
In an inertial reference frame (say you're standing on the ground looking at the Merry-go-round) and you are staring at person B on the Merry-go-round, you will see Person B moving in a circular motion. From Newton's first law, we know objects with no net force acting on them travel at a constant speed in a straight line which indicates that there is a net force on Person B (static friction acting towards the centre).
static friction opposes motion when some outer force tries to move it...
No, it is truer to say "Static friction always opposes relative motion at the point of contact". This does not necessitate the existence of another force. The centripetal force (static friction) is keeping person B constrained to a circle but is also giving them radial speed. If the force of static friction stopped, they would fly off in a tangential direction.
Person B "wants" to move in a tangential direction to the circular path because their body's inertia "wants" to keep them moving in a straight line, but as the Merry-go-round is rotating, friction will stop them moving in a straight line and act inwards. So in an inertial frame, it isn't a case of force vs force but force vs inertia.
What may confuse you is your interpretation of Newton's 3rd Law. While the force of friction acts on person B, you may think that this law states that they are constrained to the circle because of another force. But this law states that each force acts on a different object. The force of friction acts from the Merry-go-round on Person B but Person B exerts an equal and opposite force in the Merry-go-round itself, so it has nothing to do with person B being constrained to a circle. There is still a net force acting inward, but it has to do with keeping them in that circle. If you increased the centripetal force, the circle would decrease in radius.
If the forces were balanced in the inertial frame, then the net force would be $0$ and according to Newton's first law, Person B would travel in a straight line but they move in a circle.
In the rotating reference frame of the merry-go-round, we don't see Person B move in a circular motion as the reference frame moves with them, but we know the centripetal force is acting inwards so to explain the fact that the net force appears to be $0$ in the rotating reference frame, we "invent" a force (the centrifugal force). This force doesn't exist, in reality it's just the inertia which is why it's often called an inertial force.
With the merry-go-round rotating at constant angular velocity, for an object fixed at the edge of the merry-go-round, in the inertial frame (observer on the ground) the object is accelerating due to the change in its velocity vector even though its speed is constant (acceleration is a vector). The force that causes the acceleration is radially inward and is called a centripetal (center seeking) force; in this case the centripetal force is due to friction.
Viewed from a non-inertial reference frame fixed to the object, in that frame the object is at rest and the centripetal force inward due to friction is countered by a centrifugal (center fleeing) force outward. A centrifugal force only appears in a non-inertial reference frame. Other forces can also appear in a non-inertial reference frame, such as the Coriolis force that appears if the object is moving in the non-inertial frame. The forces that only appear in a non-inertial reference frame are called fictitious forces because they are not true forces in the inertial frame, but only appear in the non-inertial frame due to the acceleration of the non-inertial frame.
See a good physics text such as one by Halliday and Resnick for a basic discussion of centripetal and centrifugal forces. For more details on non-inertial reference frames and the fictitious forces that appear therein, see a good physics mechanics text such as Mechanics by Symon.
Short and simple: friction opposes relative motion between surfaces. You do not need an opposing force for there to be friction.
As a simpler example, consider the case of a block stacked on another block. You gently push on only the bottom block so that the system starts moving as a whole. The static friction acting on the top block is the only force acting on the top block; no additional force is needed to act on the top block for the friction to occur.
So, there should be some force acting radially out to cause static friction acting radially in, but even in that case sum of forces acting in radial direction is zero since static friction balances outer force trying to move us. If so, there is no net force acting radially in to allow rotation. Where have I gone wrong?
There is a force acting radially out. But as can be seen in the free body diagrams below of an object and the merry-go-round, it is the friction force the object exerts on the merry-go-round in opposition to the friction force (centripetal force) that the merry-go-round exerts on the object, as an action-reaction pair at the surface per Newton's third law. These equal and opposite forces are the reason there is no relative motion between the block and merry-go-round.
However, as shown in the FBD of the block, the friction force the merry-go-round exerts on the object is the only external horizontal force acting on the object and is analogous to the example given in the answer by @BioPhycisist. It is this net force on the object that causes it to accelerate radially inward giving it circular motion.
Hope this helps.