It is possible for an acid to burn a hole through the skin of the hand. With enough acid and enough time, you could burn through the hand itself.
In chemistry class, our teacher demonstrated how some concentrated sulfuric acid could turn a beaker of sugar (a carbohydrate, not a protein, but the closest he could get to human flesh on short notice) into a column of smoking carbon.
This piqued my interest, and I put a drop of the sulfuric acid on my own skin, while standing next to the water faucet (stupid, but not recklessly so). After several seconds I felt a mild itching sensation and rinsed. Your results may vary depending on dryness of skin and other factors – don’t tell the E.R. people I suggested this in any case.
I have seen pictures of serious acid burns in a medical textbook.
Whether the acid in carrots can be brougth to dangerous strength is beyond my ken.