Too much of a just okay thing?

Can you turn orange from eating too many carrots?

Yep.  That beta-carotene can pigment your skin just like it does your carrot cake.

How many carrots would I need to eat?

Partially depends on how bioavailable the carrots are.

As in, are you eating them raw? (more carrots needed)

juiced? (fewer carrots needed).  

Also, since beta-carotene, aka vitamin A, is fat-soluble, the less body fat you have the more likely you are to store this orange-molecule in your skin.

Benefits of eating carrots to the point of turning orange?

Sweet tan bro!

Disadvantages of eating carrots to the point of turning orange?

You probably have an eating disorder and should talk to a doctor.  Best of luck :(