I have a 14' X 6' Garden area, would watermelons ruin my other plants?

My little one wants to grow some watermelons in our garden. I have a good amount of space but I have seen some watermelon plants that were out of control. Would a small melon plant be ok? I forget the name of the watermelon plant at the nursery but the tag said they were small but good melons. Sweet Baby or Baby Red something like that.