Can there be tension in an inextensible string? 
In physics, tension describes the pulling force exerted by each end of a string, cable, chain, or similar one-dimensional continuous object, or by each end of a rod, truss member, or similar three-dimensional object. Tension is the opposite of compression.
At the atomic level, atoms or molecules have electrostatic attraction; when atoms or molecules are pulled apart from each other to gain electromagnetic potential energy, tension is produced. Each end of a string or rod under tension will pull on the object it is attached to, to restore the string/rod to its relaxed length...

So tension is caused due to the displacement of atoms with respect to each other. So strictly speaking, is it possible then to have tension in an inextensible string? 
 A: Yes.
Tension is simply the force a string exerts against you if you pull on it. An inextensible string means no matter how much you pull on it, it won't extend: it must always exert enough tension to counterbalance whatever force you use to pull on it.
A: You are confusing an ideal, mathematical model with a real physical object. They are not the same. 
An inextensible string is an ideal model, an abstract concept. It transmits any amount of tension from one point to another depending on the conditions, and it does not extend at all. Usually it has no mass and no thickness. How it manages to have such properties is irrelevant. Many real materials might behave like an inextensible string - to a good approximation. 
Real strings always extend to some extent, either because of the extension of atomic bonds, or because of an alteration in the structure (eg like a coiled spring). Real strings have mass and break when the tension exceeds a certain limit. They might also have a small amount of compressive strength, especially when the length is comparable with the diameter. They might expand or contract when heated, and have other properties (colour, hardness, roughness etc). 
Ideal strings only have the few ideal properties that are relevant to the problem at hand.
A: An inextensible string is just the limiting case of an extensible string that has a very high elastic modulus (Young's modulus).  This is very similar to @freecharly 's answer, although it was not quite correct for him to say that an ideal string is perfectly rigid, since, if that were the case, the string could not bend/flex.  So, for a real string that approaches inextensible behavior while, at the same time, being very flexible, the Young's modulus must be very high while the cross sectional moment of inertia must be very small (which allows the string to easily bend/flex).  Irrespective of whether the string is extensible or inextensible, if the string is essentially massless and friction is not acting to cause its tension to vary along its length, the tension at all locations along the length of the string must be constant.
A: Tensile stress is force per unit area. If your string cannot be elongated by this tension (doesn't have an extension strain) because of an infinite Young's modulus, it behaves like a perfectly rigid body and the tensile stress (force per unit area) you apply on one end appears also on the other end.  
