When does the story of One Piece take place? Since the story is pirate-themed, I thought it may be at the time of the Golden Age of Piracy. The Golden Age of Piracy spans from 1650s to the 1730s. So I thought it happened around that time. Am I right or do they have their own timeline?
You had a pretty close guess there.
Using Noland's Log book as a guide, it is possible to work out that the current storyline of One Piece takes place in the sixteenth century with the Kaienreki reference.
A pretty accurate world time line has also been made on the One piece wiki although only updated till November last year ('13). For more info and their sources of the years mentioned you can check out their chat/change logs here
Well, actually! One Piece doesn't even take place on earth. It's pretty obvious when you take a glance at the planet's geography:
Not even during earth's theorized pangaea phase did it ever look like this. So One Piece is either very, very, very far into the future, or more likely simply not on earth at all.
A lot of ideas in OP are gathered from the eras you've mentioned, certainly, as a thematic thing, but they don't actually take place in those points of human history.
One Piece takes place in an alternate world and within an alternate timeline.
There is at least one calendar named Kaienreki. Noland Montblanc visits Jaya during Kaienreki 1120s. That being said to happen around 400 years before current time, we can say the main story happens during Kaienreki 15XXs
I think it's far into the future on Earth and information about the past is misconstrued and only some things were carried on throughout the years like technological ideas, values, traditions, myths, phrases, etc. I also think that there was some sort of apocalypse hence the devil fruits, Giants, fishmen, angels on sky island, and the part about Cerberus during thriller bark.
The events leading up to the blank era were basically the rebuilding of humanity but much was forgotten after the apocalypse, and a group of people worked to recover that lost history and were successful. The government that was created soon after covered up those 100 years of revival because of fear that humanity would have to endure another apocalypse due to unknown reasons. The people who found out about the past before the apocalypse created the poneglyph to be indestructible using information found in the past in hopes of a full recovery for humanity.
My hunch also grew when Sanji asked Brooke if he ever heard of TMI, which is an acronym typically used in texting. Plus the Earth looks like it does due to geographical changes and the water between heaven and hell.
I believe it takes place during the 19th century, seeing as how the ships match up to those during the 1800s, the first gas operated stove was made during the 1820s and Sanji smokes cigarettes, which were invented in 1865 and mass produced across the world in 1881. This is saying IF one piece took place on Earth.
In Episode 184, one of the Shandorans says "What on Earth is going on? Those dark clouds.. What happened?" It also doesn't hurt that the dirt is called Vearth.
Its the Hulu translation, so I don't know how correct this is...(https://i.stack.imgur.com/wW9fZ.jpg)