Where can I look up the 3D positions of the closest stars?
You can determine the 3D position for a star, $p$, by using the distance from Earth, $R$, as well as the right ascension, $\Omega$, and declination, $\delta$, within the following formula.

$p = R \begin{bmatrix} \cos \Omega \cos \delta \\ \sin \Omega \cos \delta \\ \sin \delta \\ \end{bmatrix}$

Note this will yield positions within the Earth-centered inertial reference frame.