Some four-positions
We consider two points in the frame $S'$ which is moving relative to the frame $S$ at speed $v$. In the coordinates of $S'$, these points have four-positions
\begin{equation}
\mathbf{x}_A' = (t_A',x_A')^\intercal, \quad \mathbf{x}_B' = (t_B',x_B')^\intercal.
\end{equation}
The proper distance between these points is given by
\begin{equation}
\Delta s^2 = -\Delta t^{\prime 2} + \Delta x^{\prime 2}.
\end{equation}
Expanding the primed coordinates in terms of the unprimed coordinates, using the Lorentz transformations, we have
\begin{equation}
-(t_A'-t_B')^2 + (x'_A-x'_B)^2 = -(\gamma(t_A-vx_A)-\gamma(t_B-vx_B))^2 + (\gamma(x_A - v t_A) - \gamma(x_B - vt_B))^2,
\end{equation}
where we have deliberately not expanded out the brackets, so that we recall that the left squares and right squares correspond on both sides of the equation.
Bell's spaceships
Now we can state the Bell spaceship problem as follows. Consider two spaceships that are uniformly accelerated in $S$ so that the distance between them in $S$ remains constant. Between them lies a string that just spans the distance. The question is whether the string will break due to this acceleration.
Mathematically we return to our four-position analysis. We wish to define a length $L' := x_A'-x_B'$ in $S'$ and so we are required to set $t_A'=t_B'$. This cancels the left-hand square on both sides of the equation. To define a length in $S$ we must do the same, defining $L := x_A - x_B$ at $t_A = t_B$. Crucially, we now set $L$ to be constant. This implies that $L'$, i.e. the distance that the observer in $S'$ experiences, as $L' = \gamma L$. We see that the distance between these two points from the point of view of $S'$ increases as the velocity increases. Note that there is no rod here, just two-comoving points and two perspectives on their absolute distance. To see length contraction, we simply fix $L'$ as the proper distance. This means that $L$ is going to see the rod obeying $L = L'/\gamma$.
Discussion
The solution to the paradox comes when we realise that fixing the distance $L$ is going to cause real stress on the object when it wants to contract, at least from the point of view of $S$. This fixing in $S$ would be viewed in $S'$ as the gradual distancing of the two spaceships as the they accelerate, causing equal stress and equally causing the string to break. The entire paradox does not crucially rely on accelerating frames. The hidden statement of the paradox is in the rewording of the setup:
What happens to a rod if we boost it to a velocity without allowing it to contract?
Bell found that many physicists did not have good intuition for this situation and put it down to the fact that they had never been taught what I call the 'ether intuition'. That is to say that relativity can be consistently understood by invoking an ether which is a preferred state of rest but entirely undetectable. This ether imposes physical length contraction that can have physical effects like breaking the string. Both interpretations of relativity provide the same result but Bell argued that the former could provide pedagogical power.
