If you use the direct selection tool, you can drag over the point where two lines meet. Then go to Object > Path > Join (or learn the shortcut key). Repeat for each point until you have a closed shape that you can color. It may be quite time consuming though, but it depends on the shape.
If you give each line a fill anyway, and start joining a few points, you might get away without having to join all of them. If it looks filled in, you should be able to do a pathfinder operation to finish the job. Another option might be to import the artwork into Flash, which has a more organic way of working with lines and shapes. This can be re-exported later.
Shapes constructed of individual lines are not common in illustration programs, and are more likely to be from CAD applications. That's why Illustrator doesn't cater for them. It's not a failing on the part of Illustrator, although it would be nice if the pathfinder could handle them.