The generally agreed epoch of Mahabharata war (and also the start of kali yuga or iron age) among Vedic astrologers is about 3102 BC which places the epic deep into the Bronze age territory. Historians generally reject this epoch as it creates the dark age problem (over 2000 years of unexplained historical gap between that war and 500 BC when figures like Bimbisara, Buddha and Ajata Satru start appearing). Their epoch is in the range of 800-2000 BC depending on who you ask.

Having said that, there is nothing to make the bronze age epoch impossible. There is nothing in the epic which may not have been possible in the bronze age based on what we already know about history.