1. **Define the line $\ell$ and point $P$:**
Let $\ell$ be the line defined by the points $(-d, 0, z)$, and let $P = (0, 0, 0)$.

2. **Set up the inequality for the set $S$:**
The set $S$ consists of points $X = (x, y, z)$ such that the distance from $X$ to $\ell$ is at least twice the distance from $X$ to $P$. Mathematically, this can be written as:
$$\text{dist}(X, \ell) \geq 2 \cdot \text{dist}(X, P)$$

3. **Calculate the distance from $X$ to $\ell$:**
The distance from a point $(x, y, z)$ to the line $\ell$ (which is parallel to the $z$-axis and passes through $(-d, 0, 0)$) is given by:
$$\text{dist}(X, \ell) = \sqrt{(x + d)^2 + y^2}$$

4. **Calculate the distance from $X$ to $P$:**
The distance from a point $(x, y, z)$ to the point $P = (0, 0, 0)$ is:
$$\text{dist}(X, P) = \sqrt{x^2 + y^2 + z^2}$$

5. Set up the inequality:
$$\sqrt{(x + d)^2 + y^2} \geq 2 \sqrt{x^2 + y^2 + z^2}$$

6. Square both sides to eliminate the square roots:
$$(x + d)^2 + y^2 \geq 4(x^2 + y^2 + z^2)$$

7. Expand and simplify the inequality:
$$x^2 + 2dx + d^2 + y^2 \geq 4x^2 + 4y^2 + 4z^2$$
$$d^2 + 2dx \geq 3x^2 + 3y^2 + 4z^2$$
$$3x^2 + 3y^2 + 4z^2 - 2dx \leq d^2$$

8. **Complete the square for the $x$ term:**
$$3 \left( x^2 - \frac{2dx}{3} \right) + 3y^2 + 4z^2 \leq d^2$$
$$3 \left( x - \frac{d}{3} \right)^2 - \frac{d^2}{3} + 3y^2 + 4z^2 \leq d^2$$
$$3 \left( x - \frac{d}{3} \right)^2 + 3y^2 + 4z^2 \leq \frac{4d^2}{3}$$

9. Divide through by 3:
$$\left( x - \frac{d}{3} \right)^2 + y^2 + \frac{4z^2}{3} \leq \frac{4d^2}{9}$$

10. Recognize the equation of an ellipsoid:
The inequality represents an ellipsoid centered at $\left( \frac{d}{3}, 0, 0 \right)$ with semi-axes $\frac{2d}{3}$, $\frac{2d}{3}$, and $\frac{d}{\sqrt{3}}$.

11. Calculate the volume of the ellipsoid:
The volume $V$ of an ellipsoid with semi-axes $a$, $b$, and $c$ is given by:
$$V = \frac{4}{3} \pi abc$$
Here, $a = \frac{2d}{3}$, $b = \frac{2d}{3}$, and $c = \frac{d}{\sqrt{3}}$.

12. **Substitute the values of $a$, $b$, and $c$:**
$$V = \frac{4}{3} \pi \left( \frac{2d}{3} \right) \left( \frac{2d}{3} \right) \left( \frac{d}{\sqrt{3}} \right)$$
$$V = \frac{4}{3} \pi \cdot \frac{4d^2}{9} \cdot \frac{d}{\sqrt{3}}$$
$$V = \frac{4}{3} \pi \cdot \frac{4d^3}{9\sqrt{3}}$$
$$V = \frac{16 \pi d^3}{27\sqrt{3}}$$

The final answer is $\boxed{\frac{16 \pi d^3}{27\sqrt{3}}}$