There is outdoor seating. Ocracoke is very pet friendly but I don't know the Flying Melon's policy.
As Jill said, there is outdoor seating - having a pet outside is totally fine. Inside, it's a bit cramped, so having a pet that isn't a service animal isn't advised. Another heads up in case your pet is very friendly / curious / doesn't so well with other animals, there is a resident cat that hangs around the outside.
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