How was Dhirubhai Ambani able to build his enormous empire without any entrepreneurship background at all?

Dhirubhai Ambani is an Indian icon who did that way. He is a little more special than the American entrepreneurs above as the old world relies on connections & lineage a lot, lot more than the new world. 

Dhirubhai started out as a trader importing polyester fabric and exporting spices. The small dowry he got and experience he acquired working in middle east, helping him a bit to get started in this.  

Trading was not a sexy thing those days. Only uneducated people and very rich kids without much drive were dabbling in there. There was not enough competition for someone as smart and as fierce as Dhirubhai. Once he mastered polyester trading, he moved upstream to start producing fabric and downstream to directly brand this fabric to customers.

His apparel brand - Vimal - created a sensation in the 1970/80s. Vimal brought in top fashion designers to build the aesthetic appeal, focused a lot on fabric quality, made really sexy promotions and got a whole bunch of top model & cricketers to endorse. 

Then expanded the polyester business and started producing the things needed to make the polyester (as India's import rules were quite harsh) such as petrochemicals.

Besides the product innovation, he also utilized the financial markets to full extent. His company went to IPO in 1977 (when India's stock markets were very small) and really used the power of common investors. When I was young, my parents will always be chatting about how everyone should own Reliance stock - it took stock markets to the masses.

Dhirubhai had the dream to break in a market with too much of connections, had the persistance to stay through Indian government's moronic policies, had the innovative ability (both in product and finance) and finally the luck (from the dowry to being at a time when the Indian economy was exploding).
While a family background in entrepreneurship can help you be a good entrepreneur, it doesn't help a lot in being a legendary entrepreneur. Legends completely blaze their own path without being encumbered by the past.