I have looked for an answer for this question which seems trivial, but I didn't find any.
Can a Turing machine, given a word $w$, calculate the length of the word?
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–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.