Let $P_1P_2\ldots P_{24}$ be a regular 24-sided polygon inscribed in a circle $\omega$ with circumference 24. We aim to determine the number of ways to choose sets of eight distinct vertices from these 24 such that none of the arcs has length 3 or 8.

We generalize the problem by considering a regular polygon with $3n$ vertices and selecting $n$ vertices such that no two selected vertices are 3 or $n$ apart. Label the vertices $1, 2, \ldots, 3n$ and group them into sets of three: $\{1, n+1, 2n+1\}$, $\{2, n+2, 2n+2\}$, and so on until $\{n, 2n, 3n\}$. Since we need to select $n$ vertices, one from each group, the condition that no two vertices are $n$ apart is automatically satisfied.

Next, we need to ensure that no two selected vertices are 3 apart. Let $a_n$ denote the number of ways to select $n$ vertices with the given properties. Clearly, $a_1 = 0$ because each vertex is three apart from itself. For $a_2$, we manually compute that there are 6 valid sets.

To find a general formula, we construct a recursion relation. Initially, there are $3 \cdot 2^{n-1}$ ways to select $n$ vertices, ignoring the condition that no two vertices can be 3 apart. However, this count overestimates the number of valid sets. The overcount is equal to the number of valid sets of $n-1$ vertices, leading to the recursion relation:
$$a_n = 3 \cdot 2^{n-1} - a_{n-1}.$$

To solve this, we derive a closed form. From the recursion relation, we get:
$$a_{n+1} = 3 \cdot 2^n - a_n.$$
Subtracting the first equation from the second and simplifying, we obtain:
$$a_{n+1} = 3 \cdot 2^{n-1} + a_{n-1}.$$
Further manipulation yields:
$$a_n - a_{n+1} = -2a_{n-1}.$$
Rearranging and shifting indices, we find:
$$a_n = a_{n-1} + 2a_{n-2}.$$

The characteristic polynomial of this recurrence relation has roots 2 and -1, giving us the general solution:
$$a_n = A \cdot 2^n + B(-1)^n.$$
Using the initial conditions $a_1 = 0$ and $a_2 = 6$, we determine the constants $A$ and $B$:
$$A = 1, \quad B = 2.$$
Thus, the closed form is:
$$a_n = 2^n + 2(-1)^n.$$

For $n = 8$, we have:
$$a_8 = 2^8 + 2(-1)^8 = 256 + 2 = 258.$$

The answer is: \boxed{258}.