Spiders, being arachnids, do not have an internal skeleton, so no, they don't have bones. What they do have is a rigid exoskeleton, composed of a polysaccharide called chitin. This type of exoskeleton is shared between all insects and arachnids, and can even be traced back to trilobites in the Cambrian era. It's an incredibly ancient structure, but it's one that works.
And no, to satisfy your curiosity, spider silk is not also composed of chitin. It's a protein fibre. I just thought you might ask that.