Can any non-halting Turing Machines with blank tape input print out any irrational number?
I choose to represent numbers in base $\pi$. My machine prints $10$ and terminates, that's exactly $\pi$.

A Turing machine performs meaningless operations on meaningless objects, you provide the semantics, you find meaning in the computation. The answer to your question is a direct consequence of how you decide to represent numbers with the finite alphabet of the machine. It is certainly true that not all real numbers can be exactly computed under an unified semantics by Turing machines: there are uncountably many real numbers but countably many Turing machines.

But you might be able to represent some of them. For instance, with enough symbols to write integers, fractions, square roots and the imaginary unit, you can represent exactly the segments constructible with straightedge and compass, each of them in a finite amount of space.