Can we prove that a statement cannot be proved?

From the time of Euclid on, there was interest in showing that Euclid's Fifth Postulate follows from the rest of Euclid's axioms. (Roughly speaking, the Fifth Postulate says that through a given point there is a unique line parallel to a given line.)

Finally, in the $1830$'s, Bolyai and Lobachevsky independently showed that the Fifth Postulate does not follow from the rest, by discovering hyperbolic geometry. To use modern language, they found a model of the remaining axioms in which the Parallel Postulate fails.

The work of Bolyai and Lobachevsky is, arguably, the first independence result. It may have helped to change the notion of what one means by an axiomatic system.

Yes. Gödel's incompleteness theorem tells us that if a theory (collection of axioms) is "simple enough" and it can describe elementary arithmetic, then it cannot prove or disprove everything, unless it is inconsistent.

Where simple enough means that we can write a computer program which will tell us whether or not something is an axiom in our theory or not.

In addition Gödel's completeness theorem tells us that if a theory is consistent then it has a model. If we have a theory $T$ and we can find a model of $T$ where $\varphi$ holds, and another model where $\lnot\varphi$ holds, then we have proved that we cannot prove $\varphi$ from the axioms of $T$.

Such method was used to show that the continuum hypothesis cannot be proved from the axioms of ZFC; and that the axiom of choice cannot be proved nor disproved from the axioms of ZF.

One simpler example for this is that you cannot prove solely from the properties of a field that there exists a square root for the number $2$. In the field $\mathbb Q$ such number does not exist, whereas in $\mathbb R$ it does.

Another simple example. In the theory of groups, it is impossible to prove the commutative law. To see this, we exhibit a non-commutative group.

In your example, it probably is technically senseless to say "the Riemann Hypothesis is not provable". Instead you include the axiom system you have in mind, for example maybe you want to say "the Riemann Hypothesis is unprovabie from the Zermelo-Frenkel axioms of set theory".