5. $n=1,2,3$.

Let $X^{\prime}$ denote the gas station symmetric to $X$ with respect to the center of the sphere.
When $n=1$, the conclusion is obviously true.
For $n=2$, let $A B=A^{\prime} B^{\prime}$ be the minimum spherical distance between two gas stations.

By the given condition, a car starting from $A$ can reach $A^{\prime}$. Assume this path is $A \rightarrow B^{\prime} \rightarrow A^{\prime}$ (the path $A \rightarrow B \rightarrow A^{\prime}$ is similar). The fuel at $B$ is sufficient for the car to travel to the nearest gas station $A$, thus, $B \rightarrow A \rightarrow B^{\prime} \rightarrow A^{\prime}$ is a feasible path.

For $n=3$, let the six gas stations be $A, A^{\prime}, B, B^{\prime}, C, C^{\prime}$. As shown in Figure 3, let $A B=A^{\prime} B^{\prime}$ be the minimum spherical distance between two gas stations, and assume $B$ is the nearest gas station to $C$. Divide all gas stations into two groups:
$$S=\{A, B, C\}, S^{\prime}=\left\{A^{\prime}, B^{\prime}, C^{\prime}\right\},$$

A car starting from any gas station can reach the other group.
(1) If a car starting from $A$ cannot reach $S^{\prime}$ via the path $A \rightarrow B$, then it also cannot reach $S^{\prime}$ via the path $A \rightarrow C$, otherwise, the path $A \rightarrow B \rightarrow C \rightarrow S^{\prime}$ would be feasible (since the fuel at $B$ is not less than the consumption of the path $A \rightarrow B$, and $B C \leqslant A C$). Therefore, a car starting from $A$ can only directly reach $S^{\prime}$. The nearest gas station in $S^{\prime}$ to $A$ is $C^{\prime}$, thus, the path $C \rightarrow B \rightarrow A \rightarrow C^{\prime} \rightarrow B^{\prime} \rightarrow A^{\prime}$ is feasible.

Similarly, if a car starting from $A^{\prime}$ cannot reach $S$ via the path $A^{\prime} \rightarrow B^{\prime}$, then the path $C^{\prime} \rightarrow B^{\prime} \rightarrow A^{\prime} \rightarrow C \rightarrow B \rightarrow A$ is feasible.
(2) If the assumption in (1) does not hold, then the car can first travel directly from $A$ to $B$. Since $S^{\prime}$ is still reachable, and $B C \leqslant B C^{\prime}=d\left(B, S^{\prime}\right)$, the car can continue from $B$ to $C$.
The fuel at $C$ is not less than the consumption of the path $B \rightarrow C$, and
$$d\left(C, S^{\prime}\right)=C A^{\prime} \leqslant B C^{\prime}=d\left(B, S^{\prime}\right),$$

Thus, after reaching $C$, $S^{\prime}$ is still reachable, and the nearest gas station in $S^{\prime}$ to $C$ is $A^{\prime}$.

Therefore, the path $A \rightarrow B \rightarrow C \rightarrow A^{\prime} \rightarrow B^{\prime} \rightarrow C^{\prime}$ is feasible.
The above proves that $n=1,2,3$ all satisfy the requirements.
Next, construct a counterexample for $n \geqslant 4$.
As shown in Figure 4, assume the half-circumference of a great circle on the planet is 1.

Place the gas stations $A_{2}, A_{3}, \cdots, A_{n}$ on a great circle, and satisfy
$$A_{2} A_{3}=A_{3} A_{4}=\cdots=A_{n-1} A_{n}=d<\frac{1}{n-1},$$

and arrange the position of gas station $A_{1}$ such that
$$A_{1} A_{3}=d \text {, and } A_{1} A_{2}=A_{1} A_{4} \text {. }$$

Assume the gas stations $A_{1}, A_{1}^{\prime}, \cdots, A_{n-1}, A_{n-1}^{\prime}$ each store the fuel required to travel a distance $d$, and $A_{n} 、 A_{n}^{\prime}$ each store the fuel required to travel a distance $1-(n-2) d$. Then a car starting from any gas station can reach the symmetric gas station. The corresponding feasible paths are
$$\begin{array}{ll} & A_{1} \rightarrow A_{3} \rightarrow A_{4} \rightarrow \cdots \rightarrow A_{n} \rightarrow A_{2}^{\prime} \rightarrow A_{3}^{\prime} \rightarrow A_{1}^{\prime} \\ \text { and } & A_{i} \rightarrow A_{i+1} \rightarrow \cdots \rightarrow A_{n} \rightarrow A_{2}^{\prime} \rightarrow A_{3}^{\prime} \rightarrow \cdots \rightarrow A_{i}^{\prime}, \end{array}$$

where $2 \leqslant i \leqslant n$.
However, let
$$S=\left\{A_{1}, A_{2}, \cdots, A_{n}\right\}, S^{\prime}=\left\{A_{1}^{\prime}, A_{2}^{\prime}, \cdots, A_{n}^{\prime}\right\},$$
then the fuel at each of $A_{1}, A_{2}, \cdots, A_{n-1}$ is only sufficient for the car to travel to the nearest gas station, and the fuel at $A_{n} 、 A_{n}^{\prime}$ is only sufficient for the car to travel to the other group ($A_{n}$ to $A_{2}^{\prime}$, $A_{n}^{\prime}$ to $A_{2}$). Therefore, to visit all gas stations, a car starting from $S$ can only visit all gas stations in $S$ without using the fuel at $A_{n}$, which is impossible.
A car starting from $S^{\prime}$ is similar.
This provides a counterexample for $n \geqslant 4$.
In summary, the values of $n$ that satisfy the requirements are only $1, 2, 3$.