The empty set contains irrationals only. Is the above statement true; because we can not find any rational in it.
Or false; because it does not contain any irrational.
Also is the statement: "The elements of the empty set are irrationals and no element in it is rational." true, being vacuous?
 A: Yes the statement is true. It can be rephrased as if $x \in \emptyset$ then $x \in \mathbb Q^c$. Since the hypothesis $x\in \emptyset$ is false the statement as a whole is by definition true.
Another way to think about this is that the statement "if $x \in \emptyset$ then $x \in \mathbb Q^c$" is that same as saying that $\emptyset$ is a subset of $\mathbb Q^c$ which is also vacuously true.
A: We say that a set $A$ has only irrational numbers if $A$ is a subset of $\Bbb{R\setminus Q}$.
The empty set is a subset of every set. Therefore it is indeed correct to say that it only has irrational numbers as elements. It is also true to say that it has pink elephants as elements, as well as almond milk and tofu cheese, or snarks and grumplings.
Another way of looking at this is to say that $A\subseteq\Bbb R$ has only irrational numbers if $A\cap\Bbb Q$ is empty, i.e. if $A$ is disjoint from $\Bbb Q$. And again, $\varnothing\cap\Bbb Q$ is indeed empty.
A: Yes: $$\forall x \in \emptyset: x \text{ is irrational}$$ 
holds but also
$$\forall x \in \emptyset: x \text{ is rational}$$ 
holds as well. Proof: if it doesn't hold, there should be a counterexample.
But there is no element in $\emptyset$ to be a counterexample.
Read wikipedia on vacuous truth, e.g.
