This question prompts me to evaluate the possibility that the romans or greeks could be blown in a storm to the Americas and survive if they had enough water provisions?
Or maybe not provisions but expertise of some sort to survive the long journey? Is this possible (even if very remotely).
Just a reality check.
This is possible, but rather improbable.
Ancient vessels were very lightly constructed, the usual practice to build the shell of the boat first and the insert ribs and framing, and warships like a trireme are actually very lightly built, so the oarsmen can bring it up to speed rapidly, execute complex manoeuvres and so on. A large storm would be a grave danger to such a ship, and at the first sign of heavy weather the captain wold most likely order the ship to be beached in order to be safe. Since ancient ships also spent most of their time sailing close to shore, and generally beached at night, it would be very difficult for a storm to really pul a ship out to sea. If a ship was to be caught in a storm, then the most likely outcome would be the ship would break up and sink.
Triremes had the added disadvantage of having large crews packed in a very small space, so there was not a lot of on board supplies. Water, in particular, would be a huge issue, and a human being will die after three days without water. A trireme would have over 100 people on board, and contrary to the movies, these were all highly skilled crew members, not slaves (a slave on the oars of a tightly packed rowing bench could cause disruption by not being able or willing to keep up with the ever changing rhythm of the oars as the ship manoeuvred). Throwing the oarsmen overboard to conserve supplies would be problematic, especially since they outnumbered the rest of the crew by a very large margin. A ship adrift under these conditions would probably see a large percentage of the crew die due to thirst, or in a bloody mutiny.
A more probable scenario is a "round ship" carrying cargo and a small crew of professional sailors being blown off course and ending up in the Americas. Since the ship is larger and more stoutly built, it is more likely to survive a storm, and since the crew is small, they can make the supplies last longer (and indeed they may be able to pillage the cargo for extra supplies). Once ashore, they also have the advantages of having trade goods to establish communications with the natives, and sailors in those days would normally also be traders, so able to make sharp deals once ashore.
If they were in the Mediterranean (their usual territory) then I would say not without divine intervention.
If they were in the Eastern Atlantic, perhaps off the coast of Spain or Morocco, then I think it could happen. People have survived adrift for very long periods. Steven Callahan survived for 76 days adrift in a life raft. Richard Van Pham survived for three and a half months adrift in a sail boat without a sail. That's more than long enough to cross the Atlantic: 3000 miles at 3 knots would take about 40 days. A boat with a crew would have even better prospects with more options (resorting to cannibalism for example).
The boat and crew would be extremely lucky to survive it but it would not be impossible. They would probably not be in good condition if they did.
They would most likely arrive on the north coast of South America or the Caribbean because of the usual winds and currents. In the North Atlantic, winds tend to blow west to east but the trade winds near the equator are usually east to west.
(Picture from http://science.kennesaw.edu/~jdirnber/oceanography/)
In 1969 Thor Heyerdahl crossed the Atlantic in a papyrus boat similar to one he found on Egyptian pyramids. He was attempting to prove that Egyptians could have colonized the Americas.
This is impossible, but not because a journey of that length is unsurvivable.
The distance between the Mediterranean and the Americas is something like 4,000 to 5,500 miles (depending on your landing point).
First: no storm can span that distance, which means it is impossible for the storm to simply deposit the boat near the Americas, deus ex machina style. This brings us to the second problem.
Second: the Americas lie West of the Mediterranean, while Greece is toward the Eastern side of the Mediterranean. East and West can readily be distinguished by the rising and setting of the sun. It is unthinkable that the crew of a shore-hugging boat would head directly away from their home port for days or even weeks after being struck by a very powerful storm, and it is similarly impossible for them to be mistaken about whether they are heading East or West. Unless every one of the crew is insane, they will immediately try to reach the shore so they can take on food and fresh water, and (likely) repair their boat. The boat will probably head directly Northeast until it sights shore.
So, while a Greek or Roman vessel crewed by contemporary sailors may be capable of crossing the Atlantic (as other answers describe), it is utterly impossible for this to happen by accident or force. For this scenario to take place, it is an absolute necessity that the sailors actively undertake most of the journey deliberately.
