Absolutely not. Of course it is vital if you want to be a doctor or lawyer, but for those who would rather pursue a life of entrepreneurship a degree is not necessarily a requirement. In fact the argument could be made that a young entrepreneur without the burden of college loans to pay back is actually in a more favorable position to find eventual success.
How do you define successful? There are many examples of multimillionaires who never received a degree. It depends on how focused you are and what kind of ideas you have. If you want to seek certain professional careers then it can become necessary in order to gain licensure, but otherwise you can go far on smarts and hard work, with the occasional good idea to get you going.