This is a great question and worth asking. In order to fully answer it, we should look at the ancient greek word that's used to describe Jesus' occupation. In most ancient manuscripts, for instance P45, they use the ancient Greek word "tekton," which is usually translated as “carpenter" although there are other professions it could have been.

To quote the New Testament historian Bart Ehrman, he said:

The word could refer to a number of occupations that involved working with the hands—stone mason or metal worker, for example. In any event, a tekton was a lower-class, blue-collar worker. If Joseph worked with wood, it would have been to make things like plows, yokes, and gates, not fine cabinetry." Ehrman, Bart D. Jesus: Apocalyptic Prophet of the New Millennium (p. 99). Oxford University Press)

I would also add in that the idea of him making tables or chairs to be highly unlikely as well.
But it's also worth noting that tekton is not the only word used to describe Jesus and his father's occupation. Another historian which discusses this is John Dominic Crossan in his book The Historical Jesus: The Life of a Mediterranean Jewish Peasant whom wrote:

Those terms include “weaver,” both of wool (eriourgos) and linen (linourgos), and “carpenter” (tektön), the same term used for Jesus in Mark 6:3 and for Joseph in Matthew 13:55. (Crossan, John Dominic (2010-06-30). The Historical Jesus: The Life of a Mediterranean Jewish Peasant (p. 29). HarperCollins.)

So long story short: Jesus could have been a carpenter, but it would have been for more industrial types of objects that he would have created. Please let me know if you have any follow up questions.

Edits: for spelling

Also, is there any veracity to him being involved with the invention of the sit down tabled, as depicted in The Passion? Or a contemporary of his?