When you have a hole in a stainless steel sink, welding will permanently repair the hole. The problem with welding is the cost of the equipment and the training required to use the equipment. These two issues generally will force you to look for another option--and there is another option: epoxy formulated with stainless steel that does not corrode and bonds readily to the surface of the stainless steel sink with minimal preparation. This creates a near permanent repair to your stainless steel sink.
Place the small wooden block inside of the stainless steel sink and slide the wooden block over the hole you need to repair.
Apply pressure on the small wooden block with one hand and hit the hole from the underside of the stainless steel sink with the wooden or rubber mallet to close the size of the hole. The smaller you can collapse the hole, the better the final repair.
Sand the underside of the sink where the hole is located to give the epoxy a good bonding surface.
Spray the 50/50 mixture onto the underside of the stainless steel sink and wipe the area with a clean rag to remove impurities and sanding dust from the underside surface.
Mix a portion of epoxy to cover the hole. Make sure that you mix the epoxy as recommended on the packaging.
Place the epoxy over the hole, on the underside of the stainless steel sink, and work the epoxy onto the surface of the sink by pushing it to the sink's surface with your thumbs. Make sure that all air has been removed from the epoxy.
Inspect the inside of the stainless steel sink to ensure that no epoxy is emitting from the hole. If you see epoxy, push it back through the hole to create a smooth surface inside of the sink.
Allow the epoxy to cure for the time specified on the packaging before using the repaired stainless steel sink.