Finding a common interval or range between different vectors or arrays

Hello community, I am facing a problem here.

I have different time segments, with different start and end times and also with different time spacing or dt (sampling rates), so let's say that none of the data points inside of each time segment are the same between data points in the other time segments, therefore intersect function might not help me.

Speaking with an image:

At the end, I just need the min and max time values of the shaded rectangle.

I am hardly trying but at this point I can't find a solution, can anybody please help me?