It's not even clear that he was a carpenter. The Greek word used is "Tekton" (τέκτων). Mark uses it to describe Jesus's job (or rather, some people from his hometown who refuse to accept Jesus's claims to spiritual authority say "Isn't he just a /tekton/?"). Matthew has almost the same, but in the story they say "isn't he just the son of a /tekton/?"

But what is the meaning of /tekton/? Generally, scholars of ancient Greek consider it to mean a craftsman or artisan, such as a carpenter or builder.

It does seem to be distinguished from a metal-worker. The Septuagint (Greek version of the Old Testament) refers to /tekton/s and smiths as two separate things, for example.

So carpenter is reasonable, but it does generally seem to be in a slightly more general sense, associated with building. So maybe not someone who builds, say, chairs, but rather one who helps build buildings.

This is a somewhat common term for artisan though, so we can't be sure exactly what it refers to. Also, Nazareth was not a very large settlement (though it was near a Greek city, Caesarea, that was being built during this period, so may have supplied labourers to help with the building), so it's quite possible that someone who, say, was primarily a builder also did other forms of craft.

I also think it's relevant to look at the kind of parables and phrases Jesus is quoted as using.

For some examples:

"Therefore everyone who hears these words of Mine and acts on them, may be compared to a wise man who built his house on the rock."

"Did you not read in the Scriptures, 'The stone which the builder rejected, has become the chief corner stone'?"

“Suppose one of you wants to build a tower. Won’t you first sit down and estimate the cost to see if you have enough money to complete it? For if you lay the foundation and are not able to finish it, everyone who sees it will ridicule you, saying, ‘This person began to build and wasn’t able to finish.’"

"Blessed are you, Simon son of Jonah, for this was not revealed to you by flesh and blood, but by my Father in heaven. And I tell you that you are Peter, and on this rock I will build my church, and the gates of Hades will not overcome it."

Jesus seems to use a number of building references, yet there are no parables about a carpenter, or parables that refer to, say, building a wooden chair or bed.

All this seems to imply that Jesus was involved in building, but it's not exactly concrete. The number of references to his job is too small and vague to be sure. Myself, I think it's most likely he was a builder, but the evidence is too thin to make confident statements.