No. Spiders and crabs are both arthropods—animals with a hard, jointed exoskeleton and segmented bodies—but they belong to different major groups within that phylum. Spiders are arachnids in the subphylum Chelicerata; crabs are crustaceans in the subphylum Crustacea.
Spiders typically have four pairs of walking legs and feeding appendages called chelicerae, and most species live on land. Crabs are chiefly aquatic crustaceans distinguished by two pairs of antennae and jaw-like mouthparts; many large crabs and shrimps are common bottom-dwelling marine arthropods. Sharing the arthropod body plan does not make spiders a kind of crab (nor crabs a kind of spider).