Computability - Can a Turing Machine calculate the input's length?
Yes, of course it can. As a simple intuition, replace "Turing machine" with "computer program" and the answer shouldn't change, assuming that the Church&ndash;Turing thesis is true.

Just use a portion of the tape after the input as a binary counter and shuttle back and forth increasing that counter by one for every character in the input, marking the characters that you've already counted.