Do It Yourself mentions epoxy resin as a method of fixing narrow-yet-deep holes. Sand the area around the hole with 100-grit sandpaper, and push the resin into the hole with a thin applicator. Smooth away any resin around the surface, and allow to dry as prescribed by the instructions. Then, using 200-grit sandpaper, even out the resin with the rest of the surface.
Do It Yourself further notes that fiberglass fabric should be used on wider holes. Use epoxy resin to fill the hole halfway, and insert the shape of the fiberglass fabric into the void. Use more resin on top of the fabric until it is visible. Use 200-grit sandpaper to even out the surface, cut away any fabric around the edges, and sand as normal. Use another coat of resin, and allow it to settle. Sand more to further even the surface.
Before applying resin, the tub needs to be cleaned thoroughly. A mix of equal amounts of oxygen bleach and water is a good tub-cleaning solution. Apply with a sponge; also, if wet sandpaper has been used around the hole itself to even out edges, remove the residue with a paintbrush, rinse area and allow to dry. Resin is usually white in color, but painting the repaired area is necessary if the tub is tinted; sometimes painting the entire tub is necessary. Finish the surface with wax to keep soap residue at bay. Of course, the wax should not contain chemicals that bother the skin.