Excluded Volume in van Der Waals equation
We assume that the molecules are identical hard spheres with diameter $d$. The excluded volume is emphatically not the volume of the sphere, but rather the volume excluded to the other particles by the presence of that sphere.

When two particles collide, they touch each other at some point on their surface, and the distance between the center of one molecule to that of the other is $d$. Indeed, two particles cannot get closer than a distance $d$ to each other, and hence the excluded volume of each particle is the volume of a sphere of diameter $d$—-we must take into account the radii of both colliding spheres.

Edit: This suggests that the excluded volume should be eight times as large as the original volume of a sphere, not four times. However, a collision event involves two particles, and this excluded volume is shared, in a sense, by both particles. We must therefore divide by a factor of two, and that gives the resulting multiplier of four.