I do not think it is possible, but it I'm finding it rather hard to prove rigorously.
Here is my attempt. I have scaled everything by a factor of 10 to make it all integers.

We want to cut 4 pieces from the four 12x24 rectangles such that they form a 26x30 rectangle.
We can assume that each of the four pieces is cut from a different rectangle. If it is possible for those pieces to form the goal rectangle, then we may as well not cut them out and use the four rectangles themselves to cover the goal as long as we allow them to overlap.
Consider the 8 points marked in red in this picture:
It is straightforward but tedious to prove that you can cover no more than two of them by a 12x24 rectangle. The diagonal of 12x24 is $12\sqrt{3} \approx 26.83$, so can just about cover the short edge of the goal rectangle, but there is not enough width left to include a third red point. Therefore each rectangle must cover exactly two of the points.
Now look at the 5 points marked in blue in this picture:
No rectangle can cover three blue points (and two reds). Obviously we need at least one rectangle to cover two of the blue points. There are essentially only three ways to do this.
It seems that none of these allow the remaining area to be covered by the other three rectangles. (This assertion still needs proof.)
The best I managed to get is the following:
There are two very small uncovered triangles at the arrows.