Part of this question has an ad hoc nature that, in my opinion, weakeans it as a math question. How much spaghetti can you fit into a Volkswagen Beetle? Remark: Slightly more than otherwise if you remember to open the glove compartment. Okay, the question is not quite that bad, but it is clearly sensitive to boundary behavior. If you allow the box to be an arbitrary convex shape (say), then who knows if it can ever be completely solved.
So let's look at the boundary-independent part of the question. In 2 dimensions, if you have a convex box of any shape with a very large inradius, then it is not hard to show that the asymptotic density of the rope is 1. You can go back and forth across the box in a boustrophedonic pattern (like the diagram on the left).
In 3 dimensions, one can speculate that the asymptotic density of the rope is $\sqrt{\pi}/12$, the same as the circle packing density in the plane. There is a theorem of Andras Bezdek and my dad, Włodzimierz Kuperberg, that the maximum density space packing with congruent circular cylinders of infinite length is attained when the cylinders are parallel. Their theorem includes the special case of the rope question in which most of the rope is parallel to itself. (On the other hand, it is not quite obvious that the entire theorem is a special case. Given a pile of stiff, straight spaghetti, can you always efficiently connect the ends to make one long noodle?) In any case, I asked my dad this asymptotic rope question, and in his opinion, it is an open problem. A rope seems much harder to control than straight cylinders.
Also, as far as I know, the same questions concerning either straight, round cylinders or one long rope in higher dimensions are also open.
It's my philosophy that an argument that a problem is open is a valid MO answer to a MO question.
Best Answer
Here are two sources, the first of which is the more substantive. The problem is even hard to approximate, but algorithms are available that achieve about $2\frac{1}{2} \times$ the optimal packing.