A nonwoven web material that contains an elastic component is provided. The elastic component contains a crosslinked network formed from a linear block copolymer having a monoalkenyl aromatic midblock positioned between conjugated diene endblocks (e.g., butadiene-styrene-butadiene (“B-S-B”) triblock copolymer). Prior to crosslinking, such linear block copolymers have a relatively low viscosity and thus may be readily formed into a precursor elastic component that is subsequently crosslinked to achieve the desired elastic and mechanical properties. Crosslinking is typically achieved through the formation of free radicals (unpaired electrons) that link together to form a plurality of carbon-carbon covalent bonds at the conjugated diene endblocks.