I know of the basics of how the Venetians did it: by making many stops along the coast to restock on water and food supplies. The entire Venetian empire during its prime was designed around getting ships from Venice to the Levant and back. Constantinople and Alexandria were two prime targets and more than a couple of days away by galley. The islands of Corfu, Crete, Naxos but also Cyprus were common pitstops for the Venetians galleys. The Dalmatian coast was riddled with towns, some of who owned allegiance to Venice, others who were struggling against Venice's growing influnce, so navigating back through the Adriatic was easier when it came to resupplying than the Mediterranaen sea.

As on the topic of clayware, wooden boxes and barrels, I know nothing, but I suppose that there is a vast difference in how goods were stored in ancient and medieval times :)

A great read on Venice that touches on a lot of its naval practices is Rogert Crowley's City of Fortune. It is very accessible and an easy read that requires little to no pre-knowledge.

I have just got the The Complete Roman Army by Adrian Goldsworthy in my mail and there is a chapter about the Roman navy with a short paragraph answering your first question. There could be much more than 100 rowers. Roman quinqueremes were large war ships with 5 banks of rowers at each side -- 320 crew overall plus 120 marines (maybe an exaggeration). Because there were so many men on board and there was a lot of water needed for the rowers, ancient fleets could not stay at sea for much longer than 3 days at a time. After that they would have to go to a friendly port or beach the ship to reload supplies and rest the crew. It was not just the rowing but that they could not move much while on board without endangering the ship to flip over in the worst case. Or seriously annoy the wrong people you have to spend a long time with. So to move a navy around and control a part of the sea you needed to control ports and at least parts of the coastline. That means provisions would be resupplied quite frequently. As for the quarters there is nothing in the book and that is all I have.

Greek triremes had essentially no supplies on board since they were streamlined warships and, as Thucydides and Xenophon make clear, navies had to make very frequent stops either at friendly cities or, very often, just beaching the ships and sending the men out to raid and forage.

The Battle of Aegospotami demonstrated acutely just how vulnerable this made navies. The final battle of the Peloponnesian war, Lysander essentially just watched the Athenians' food-gathering habits while they were landed, and timed an attack to take them completely by surprise while they were away from their ships.

As for sleeping, it's with some respect that Xenophon writes of a naval commander who maintained such good discipline that he had his men essentially take turns sleeping and rowing so they would keep moving. The reader gets the impression this wasn't very common. "While sailing there, he at times had the men rest and even sleep and at other times set them to the oars." (Hellenika 5.1.19)

Sources for all this: Xenophon's Hellenika primarily, Thucydides secondarily.

As others have said, a navy would basically operate by sticking close to well traveled routes that allow for frequent resupply (and you can imagine what that means when we are dealing with a large naval force traveling along coastal communities). But not every ship in the ancient world was a naval vessel (or naus longa, "long ship")--there were also trading vessels (naus oneriae "ships for cargo" basically), which were built considerably differently. For one, they were primarily sail driven and thus needed considerably less crew space. Secondly, they were much wider and had a deeper draught--a naval vessel wants a shallow draught to increase speed and maneuverability, a trading vessel wants a deeper one to increase stability. Trading vessels would of course still prefer to stay by the coast, which is true even today, but they had the cargo space to carry out long distance voyages, and in fact were often larger than early modern vessels. For example, the one Red Sea wreck I know of has a length of 33 meters, as compared to the 22 meters of the Santa Maria.

Most of the answers here speak to short, coastal voyages. How did longer voyages work, like a transatlantic crossing in say, 1600?

For warships in the Classical period, the typical trireme had 170 rowers plus sailors and marines so would be near 200 in total. For the rowers there's simply no room on that ship to do anything but row. This means they also had minimal storage so carrying sufficient supplies for feeding the crew was impossible.

Typically the ship would beach itself each night where the men would go ashore to cook, eat, and sleep. If they had supply ships then the cargo could be offloaded but often they went to a nearby town to purchase their food.

The battle that ended the Peloponnesian War had the Athenian fleet beached and much of the crew off searching for food when the Spartan fleet attacked.

Edit: Here's the website for a recreated trireme which is pretty awesome.

Mildly related, kind of speaks to how bad it must have been for a while if it was only addressed in the 1800s.

Beriberi, also known as Edo sickness, was a wasting illness that killed by either pure paralysis or exhaustion, and nearly devastated the Japanese navy during the 19th century.

Basically they ate too much rice and it caused a B1 deficiency, easily avoided by eating wheat/barley items. This article has a ton of information relating to naval diets as well as developing modern medicine practices in Japan.

https://medium.com/war-is-boring/eating-too-much-rice-almost-sank-the-japanese-navy-f985772c81a6

Many of these comments state that the boats just had to constantly make stops and refresh supplies - I imagine this would require quite a bit of money. How then did they finance this at the ground level? Were these ships just constantly filled with cash?

Wouldn't a galley with slave rowers always be in danger of a mutiny, especially in battle? Would the rowers make up a majority of the crew?

I understand that it would be difficult or impossible to coordinate a mutiny with an enemy attack, as attackers would probably just keep attacking anyone on an opposing ship, but did things like cutting the rowers chains and them jumping up to join the attackers actually happen?