Can Bose–Einstein condensate (BEC) take up no space?

Yes. An infinite number of Bosons can occupy the same state. But this does not mean that they will occupy the same location. The constraint that you applied (zero temperature and pressure) incorrectly assumes that there will cease to be any motion, so that all the particles are localised, and will not escape the container. This is incorrect, as even in zero temperature (and pressure) there will still be motion, due to quantum vacuum fluctuations. These fluctuations are a manifestation of the Uncertainty relation $Delta EDelta t geqslant frac{1}{2} hbar$.

Being in the same state does not mean being at the same position.

For example, the ground state of a square well of side length $L$ goes as $Psi propto sin(x/L)$. As you make $Lrightarrow infty$, this becomes a flat distribution of infinite extent. So, indeed, they do not take zero space.

As far as pressure is concerned, the pressure $P$ of a weakly interacting Bose-Condensed gas with interaction strength $g$ is (at $T=0$): $$ P = frac{1}{2}gn^2.$$

So a non-interacting BEC ($g=0$) will have zero pressure (at $T=0$).
An attractively interacting system in unstable as it will collapse ($P<0$), whereas a repuslively interacting system has a positive pressure so it costs energy to "stuff" particles in a box. NB at $T neq 0$ there is always thermal pressure in interacting and non-interacting systems alike.