Stress/strain within an object experiencing a single applied force If I push an object in space, will any strain/stress happen within the object, though there is only 1 force rather than a pair of opposite forces on the object?
Can be any (even if trivial) stress/strain under the effect of a single force? 
 A: Yes, any force you apply will result in acceleration of the body (F=MA). A non-rigid body will also deform. 
This is exactly what's happening when a rocket engine fires up. Depending on how large the push (force) applied and how small the object (mass), the acceleration could range from tiny to huge, as in "squish" to delicate human bodies.
A: Start by thinking about an object in a gravitational field - for example, anything on the earth's surface.
Even if the object is not moving, there is a distributed force (gravity) acting on every particle of the body. Those forces are balanced by an internal stress distribution in the body, which has nothing to do with whether the body is rigid or flexible. 
For example if the object was a rectangular box resting on a horizontal plane, the stress caused by the gravitational force (i.e. "weight") varies linearly from zero at the top to a maximum value at the base, and the stress at the base is counterbalanced by the reaction force from whatever the object is resting on.
In some situations (for example civil engineering structures like bridges or buildings) these might the most significant stresses in the structure. In other mechanical engineering situations, they may be negligible compared with other stresses and strains in the structure, and therefore they might be ignored when designing or analysing the structure's behaviour.
Now, consider your object in space with a force acting on it. The force will cause the body to accelerate. But if we think about the motion in a coordinate system attached to the body, that "acceleration" looks exactly the same as a gravitational force. If the object is a spaceship with no windows to view what was outside, there is no way for the crew inside to know whether the ship is accelerating, or whether it is sitting on the ground in a gravitational field. 
So, there is definitely a stress field in the accelerating body. The amount of strain corresponding to the stress depends on the elastic properties of the body. For a rigid body, the strains and deformations will be zero (that's what "rigid" means!) For a very flexible material, the strains may be big enough to make a large change to shape of the object.
