Is it possible for a person to eat so many carrots that one turns orange?
Beta-carotene, the pigment that gives carrots their orange colour, can accumulate in the skin if taken in excess amounts. This condition is called carotenosis. Since beta-carotene is a lipohilic (fat loving: dissolves in lipids), its absorption is facilitated by dietary fats [1].

There is one report that says that excessive consumption of beta-carotene containing fruits and vegetables can lead to carotenosis [2].

Also have a look at this Scientific American blog. 





Prince, Martin R., and Joan K. Frisoli. "Beta-carotene accumulation in serum and skin." The American journal of clinical nutrition 57.2 (1993): 175-181.
Wageesha, Nekadage Don Amal, et al. "Studies on hypercarotenemia due to excessive ingestion of carrot, pumpkin and papaw." International journal of food sciences and nutrition 62.1 (2011): 20-25.