The eye of the storm is actually the calmest part of a hurricane storm. Most of the heavy and tempestuous winds are deflected from the eye of the hurricane by the Coriolis Effect. The Coriolis Effect basically describes the deflection of air as it moves from high to low pressure in the Northern Hemisphere and the Southern Hemisphere.
So, the tenth chapter of the book refers to the peace before the storm (or before Hurricane Katrina makes landfall). Since the eye of the storm is the calmest part of the hurricane, this chapter describes the surreal, peaceful moments before the hurricane makes land. In the chapter, some of the houses have been boarded up, and the families have left; only some families, like the Batiste family, are staying put. In the woods, no animals can be seen. There is at once a deadly calm to everything as well as a sense that powerful forces will soon bear down on the family.
The "endless eye" possibly refers to the endless wait (under falsely calm conditions) for the hurricane's arrival. In this chapter, the conflict between Manny and Esch foreshadow the fierceness of the approaching storm. While the family makes preparations for the storm, they are, figuratively, in the "eye of the storm," where everything is supposedly calm. However, the calm center is flanked by the gusty winds and copious raindrops that will soon batter the land as the hurricane hits.