Jesus being a carpenter is subject to debate among historians and Christians alike. There is no actual Scriptural evidence that Jesus ever was a "carpenter", however based on the times He grew up in it is an easy inference. If He truly was a carpenter, then there may be artifacts created by him circulating somewhere. However, it's a good bet to say these artifacts have degraded over the years or "current" examples are probably counterfeit for the exact reason I stated above - we really don't know if he was an actual "carpenter" or not.
Let's look at Scripture really quick to talk about why this is:
The only references to Jesus ever being a carpenter is in Mark & Matthew.
“Isn’t this the carpenter, the son of Mary, and the brother of James, Joses, Judas, and Simon?”
“Isn’t this the carpenter’s son? Isn’t His mother called Mary, and His brothers James, Joseph, Simon, and Judas?”
It's important to remember that these two accounts in the Gospel are of the same event. Seems pretty straightforward that he was a carpenter and/or the son of a carpenter. However, translation is where we have our difficulty.
The Koine Greek word τέκτων ("tekton") is the word translated as "carpenter" in these passages. This word has a wider semantic range. It can mean "carpenter", however it's terminology is much broader to include the following:
- A new construction carpenter, like a framer
- A general construction worker
- A general craftsman
- A cabinet maker
- A Cooper (a person who makes barrels)
- Or a Shipbuilder
If I were translating Mark 6:3 or Matthew 13:55 myself, I would use the term "craftsman".
There is a whole slew of other things that contributed to us just saying "Jesus was a carpenter", ranging from tradition - I cite Justin Martyr's Dialogue with Trypho, chapter 88:
And when Jesus came to the Jordan, He was considered to be the son of Joseph the carpenter; and He appeared without comeliness, as the Scriptures declared; and He was deemed a carpenter (for He was in the habit of working as a carpenter when among men, making ploughs and yokes; by which He taught the symbols of righteousness and an active life)
So we know Jesus was at least a craftsman. So, how about his work?
Well, being that we've established that He certainly probably worked with wood, we just have to look at how wood degrades over time. This depends on many factors including the type of wood, where its stored, how it's taken care of, etc. Some woods, such as cedar or redwood, are naturally resistant to rot, and may last in the open for two or more years without any special treatment. Other types of wood, such as sycamore, decay within a year.
Many noteworthy trees found here appear to be oaks. Oak can last a very long time (there's a fallen oak in my mother's yard that's been there for 15 years and it's only now beginning to rot away), however there are parts of town here with oak posts that have been there for 100+ years with little sign of wear. We're talking about 2000 year old furniture, however. So taking all that in to account, chances are all examples of Jesus' woodwork are gone. Why? Because He was a simple Jew (to those who don't know who He is/was), and depending on who His work was sold to, it probably wasn't taken care of very well. Following His crucifixion, had word gotten around, those who though He was a blasphemer may have ruined the works or sold them for money.
In short, we just don't know. However, the chances are good that objects being claimed to have been made by Jesus are counterfeit.