Could we design a Turing machine that output every bit of $\pi$ one after another correctly with limited tape length?
If by "limited tape length", you mean having a finite amount of memory, then the answer is "no".  With a finite amount of memory, the Turing machine as a whole can be modeled as a finite state machine.  A finite state machine that doesn't terminate must eventually repeat a state and thus fall into a loop. Therefore, the output of any such machine must correspond to the binary expansion of a rational number. The binary expansion of any irrational number whatsoever will require an unbounded amount of state.