I have a polyline feature consisting of a grid. I attempted to convert this grid into a polygon using the "Feature to Polygon" geoprocessing tool, and it appears that not all lines have been carried over. I've attached two screencaps. The first is the polyline and the second is the converted polygon.
As @Midavalo suggested, there is likely a small gap in your lines. Zoom in and see if all the lines are touching. It isn't surprising that this is happening on the edges of the extent of your lines feature class because internal lines would cross each other and close the shapes. But there is likely a gap where the lines don't touch at the edges.
You can either inspect the data and fix it manually or you can try adjusting the cluster tolerance in the Feature to Polygon tool.
The minimum distance separating all feature coordinates, and the distance a coordinate can move in X, Y, or both during spatial computation. The default XY tolerance is set to 0.001 meter or its equivalent in feature units.