Halting Problem ~ Turing

Idea

Can there exist a turing machine which decides the halting problem i.e. given a Turing Machine $M$ and input $x$, would it halt and spit out output?

Let’s consider the cases where the input space excludes all the possible Turing Machines and input ~ encoding of the same Turing Machine (i.e. the input which leads to Barber kind of paradox). Will such a Turing Machine exists?