Inverse system of a box function
As pointed out in the comments, the inverse system is not realizable. However, it is possible to write down an expression for the impulse response of the inverse system. We want a system that results in a Dirac impulse when convolved with the given rectangular pulse $h(t)$. This can be done as follows. We delay the rectangular impulse by $T/2$, so it starts at $t=0$. Then we add an infinite number of delayed versions of it such that the result becomes a step function $u(t)$. Each delay is an integer multiple of $T$:

$$u(t)=\sum_{n=0}^{\infty}h(t-T/2-nT)\tag{1}$$

The system that achieves this has an impulse response given by

$$f(t)=\sum_{n=0}^{\infty}\delta(t-T/2-nT)\tag{2}$$

Now we have a step function, from which we can generate a Dirac impulse by taking the derivative. So the inverse system is given by the concatenation of the system with impulse response $f(t)$ given by $(2)$ and an ideal differentiator. The total impulse response of the inverse system is then

$$g(t)=(f\star\delta')(t)=\sum_{n=0}^{\infty}\delta'(t-T/2-nT)\tag{3}$$

where $\delta'(t)$ is the distributional derivative of the Dirac impulse.

Clearly, the system with impulse response $g(t)$ given by $(3)$ is not realizable, but as a theoretical concept it is interesting, just like other ideal systems (e.g., ideal frequency selective filters, differentiators, Hilbert transformers, etc.).