Choice B is the best answer. In the fourth paragraph, the narrator describes his reluctance to leave Sempere's bookshop: "When it was time for me to leave, I would do so dragging my feet, a weight on my soul." In this context, "weight" most nearly means burden.
Choices A, C, and D are incorrect because in the context of the narrator having to do something he doesn't want to, a "weight" he had to carry most nearly means a burden, not a bulk (choice A), force (choice C), or clout (choice D).