Beside what Mr Zak said, the only (crappy) way is to use the "bisect" tool in edit mode on each object to be "cut", using the box as reference, of course cutting in ortographic front/left/top mode. That cuts yor geometry (even without faces), adding new vertices along the cut edges, so that, later you can delete unwanted geometry outside the reference box...
The bisect tool is crappy, too, it has no way to keep straight, and is not affected by snapping, too. Again, if your cuts are not "perfect", your only hope is to use the snap tools after the "bisection", aligning the new vertices resulting near the box limits, to align them to the box vertices...
If not clear, I can make some reference image...
 here you have the whol (crappy) sequence:
I started from a simple test setup but similar to your case, I hope:
seen from the side
so if you select the grouped green shape (this is only 1 mesh I just added a group to make it green like yours and also visually show it better) in edit mode, acting from the basic side view (ie: not in perspective mode, but ortographic, since you have to cut straight lines "by hand"):
You can select all the mesh (A key) and use the mesh>bisect tool.
Once you select it, you have to drag a straight line (you can start it from anywhere, I chose the intersection of green and orange lines, of course).
in the above image I angled it a bit for you to see it better, but you have to keep it straight, and drag it until it "cuts" also the bottom green/orange cross:
as soon as you release the left mouse button (LMB) the cut is set, and your mesh (also if has no faces) is cut, and new vertices made along the cut on all edges, as you see below, from an angled view.
of course you could be not as much precise cutting by hand, but you could always use now the snap tool set as you see below
then align those vertices (all together or even one by one) to your "limits box" (the orange circle hints that the selected vertices are aligned to the enclosed vertex of the "limits box"). I dragged them on the X axis in the example, and pressed the CTRL key to "snap" them until the orange circle confirms the snapping:
after doing this for all edges needed, you just need to delete "outer" vertices:
and you get something similar to what you needed, I hope:
...phew! :) what a crappy tool, I know...