Ships at sea would have enough crew to operate the ship at all hours of the day and night as that was very important for sailing. Traditionally (starting from at least the late 16th century) the crew of a ship would usually be divided into two watches called the larboard and starboard watch (meaning left and right side of the ship, respectively) and each 24 hour period was further divided into seven watches of four hours each except for two so-called "dog watches" which were two hours each. Sometimes when there were larger crews they could be divided into three watches but this was less common, although it was popular with crews because each would have to stand watch for less time and thus have more time for sleep and leisure. Despite the name, the larboard and starboard watches had nothing to do with only watching one side of the ship. The purpose of the two watches was simply to rotate being on watch over the five four hour intervals and two two hour intervals throughout the 24 hour day.

The 8:00 PM to midnight watch was called the first watch, midnight to 4:00 AM was called the middle watch, 4:00 AM to 8:00 AM was called the morning watch, 8:00 AM to noon was called the forenoon watch, and noon to 4:00 PM was called the afternoon watch. Then from 4:00 PM to 6:00 PM was the first dog watch (sometimes called the lookout watch) and from 6:00 PM to 8:00 PM was the last dog watch (The Buccaneer's Realm, p. 233). Before the early 19th century, time aboard ships was kept track of using half-hour sandglasses which were relatively reliable but to adjust for discrepancies they would be set based on the sun at noon if that was visible. During each watch, a bell would be struck every half hour with the number of rings increased each time until it reached eight rings (or eight bells) for a regular four hour watch and four bells for a two hour "dog watch" (The Buccaneer's Realm, p. 233). So. for example, if you were sleeping below deck and you heard the bell ring five times you would know you had about an hour and a half left before it was your turn to go on watch.

The reason for the dog watches was so that the watches would naturally rotate so the same sailors were alternating different hours each day because there are seven watches and seven is an odd number. This means that each day one watch would cover 14 hours in total while the other would cover 10 hours, but that would even out over a 48 hour period and in practice all sailors would work for longer than that since all hands were usually called on deck at 6:00 AM to work all day except breaks for breakfast, lunch, and some relaxation around the time of the dog watches. Furthermore, all hands could be called up on deck at any time if the situation required it because of any emergency (Earle, pp. 70-71). So how much sleep were they getting? Assuming they only slept at night the 8 PM to midnight/4 AM to 8 AM watch could have maybe gotten 4-6 hours sleep and the midnight to 4 AM watch could have maybe gotten 6-8 hours sleep. They might have been able to nap sometimes during the day but generally they were expected to work as there was nearly always something to do, although sometimes there might have been more leisure time (Earle, pp. 74-75). If there was stormy weather that required all sailors to be working the ship almost continuously for days or weeks at a time, there was definitely a lot of sleep deprivation. Here is a passage from the diary of the late 17th century sailor Edward Barlow wherein he complains about sleep deprivation:

When we went to take our rest, we were not to lie still above four hours; and many times when it blew hard were not sure to lie one hour, yea, often we were called up before we had slept half an hour and forced to go up in the maintop or foretop to take in our topsails, half awake and half asleep, with one shoe on and the other off, not having time to put it on: always sleeping in our clothes for readiness; and in stormy weather when the ship rolled and tumbled as though some great millstone were rolling up one hill and down another, we had much ado to hold ourselves fast by the small ropes from falling by the board; and being gotten up into the tops, there we must haul and pull to make fast the sail, seeing nothing but air above us and water beneath us, and that so raging as though every wave would make a grave for us: and many times in nights so dark that we could not see one another, and blowing so hard that we could not hear one another speak, being close to one another (qtd. in Earle, p. 73).

Sometimes this could contribute to fatalities as possibly in the case of one seaman who fell from the rigging and drowned while on a stormy voyage around Cape Horn in 1719, also because his hands were too cold (Shelvocke, p. 75):

... as we were furling the mainsail, one William Camell cried out that his hands were so benumbed, he could not hold himself; but before those who were next to him, could lay hold of him, he fell into the sea, and the ship making very fresh way, and the sea running very high, we lost sight of him before we could bring to.

Tiredness from lack of sleep was sometimes cited as the cause of lookouts failing to notice obstacles or other ships or alternatively thinking they saw things that weren't actually there. For example:

"[W]e being all tired with pumping, watching and labouring all night ... were taking a little rest, leaving only one man to look out did not see the Privateer till he was too near us to avoid him" (qtd. in The Sea Rover's Practice, p. 107).

As for sleeping accommodations, sailors would usually sleep in hammocks below deck which were considered very comfortable especially in hot weather but they could also sleep in bunks or cots or just lying on whatever blankets or mats or pillows they had with them. Sometimes they would even sleep on deck in the tropics if the weather was calm or they were at anchor, although this wasn't considered healthy by some (The Sea Rover's Practice, pp. 88-89; The Buccaneer's Realm, pp. 16-17). Many sailors were known to sleep in wet clothes in their hammocks which caused them to stink in the tropics, but it suggests they were too lazy or more probably too tired to change once they had a chance to fall asleep (The Sea Rover's Practice, p. 87).

Sources:

Sailors: English Merchant Seamen 1650-1775 by Peter Earle

The Buccaneer's Realm: Pirate Life on the Spanish Main, 1674-1688 by Benerson Little

The Sea Rover's Practice: Pirate Tactics and Techniques, 1630-1730 by Benerson Little

A Voyage Round The World by Way of the Great South Sea by George Shelvocke