If it's worn through, you'll have to replace it. If it's just discolored, try Bartender's Helper. Just make a paste and gently put it on there with a gloved finger. Leave for 5 minutes and rinse. If that doesn't work, go ahead and replace. Check at local hardware stores to see if they have an epoxy paint that works.???
Try using Bar Keepers Friend . I use the powder/dry product and it works fantastic!
They can be professionally painted but it wouldn’t be a permanent solution. I would replace it. Stainless steel sinks aren’t expensive. The lower the guage the thicker the steel. They usually sell 16 and 18 guage. 16 guage is better.
How old is the sink?
Yes, there comes a time in the lifespan of these kinds of household items that are so heavily used; a time to replace it.