Large hole in porcelain kitchen sink - is it fixable?
I think for a temporary fix you could use a good two-part epoxy. I don't consider it a DIY thing to completely restore a sink like that, but there are companies that refresh bathtubs. I'm not sure what they apply would be robust enough for a sink, though. 

From the look of the nearby chips and the crack extending from the new damage, that sink has served its useful life and should be replaced. That's usually not an overly complicated project and tends to bring new life to a kitchen workspace.