Spider silk is not stronger than steel. In a review of studies on spider silk properties the strongest reported value was 1652 MPa ultimate tensile strength [1]. If you have a block of knives in your kitchen you own steel that is stronger than the strongest spider silk ever reported. In fact, the strongest steel that I am aware of was reported as 6350 MPa [2]. The study that reported that high strength value for steel was not even an attempt to set a record, but a study on the effect of different annealing treatments on high strength steel wire. They got the steel from a commercial supplier. However, articles for the public about steel being super strong are not as sexy as talking about spider silk.

[1] Agnarsson, Ingi, Matjaž Kuntner, and Todd A. Blackledge. “Bioprospecting finds the toughest biological material: extraordinary silk from a giant riverine orb spider.” PloS one 5, no. 9 (2010): e11234.

[2] Li, Y. J., P. Choi, S. Goto, C. Borchers, D. Raabe, and R. Kirchheim. “Evolution of strength and microstructure during annealing of heavily cold-drawn 6.3 GPa hypereutectoid pearlitic steel wire.” Acta Materialia 60, no. 9 (2012): 4005-4016.