In fiction (or science fiction) a captain’s log is a real narrative device. But would a real one have told us much about the actual people?
I can answer in particular for the Royal Navy of the later 18th and early 19th centuries. RN logs of this period were quite standardized administrative documents, with fairly little room for narrative flair. Logs were generally the responsibility of the sailing master, a warrant officer who handled the day-to-day minutia of running the ship, and the master's mates. These were not the only ship's logs- often the captain and lieutenants would have their own as well, but they were based primarily on that kept by the master. An officer might add additional notes like descriptions of ports visited or sketches thereof, but nothing of a personal nature.
In terms of how the logs were formatted, there were generally four columns to record:
These four data points were recorded each hour, around the clock, every day the ship was at sea. Notes were also kept of the ship's reckoned distance traveled and position at noon each day.
There was also room for "Remarks", which are more freeform, but these were still quite dry administrative details. By far the majority were comments on the weather and operation of the ship, such as taking in and out of sails. Sightings of other ships were invariably noted, as were landmarks and the bearings to them (these were important for determining one's position). Opening of casks of supplies, taking in fresh produce, punishment of sailors, mustering the ship's company (a weekly occurrence), and other details were included as well. Even the death of a crewmember would be noted with the same calm, factual tone; "Departed this life William Bates" or "Departed this life Thomas George, seaman".
I can offer some examples, thanks to the many digitized logs that the UK National Archives has made available. Transcribed to the best of my ability, here is a typical day in the life for three separate Royal Navy ships.
HM ship Endeavour on Tuesday, August 30th, 1768:
Moderate & Cloudy Weather. saw a sail to the southward
half past one Brought to & spoke Her being his Majesty's Ship the Guadaloupe Capt. Cornwallis from Gibraltar bound to England. At 2 made sail.
Fresh Gales & Cloudy - In 2 Reefs of the Tops[ails]
Squally. Got down Topgallant yards & Handed the Miz[zen] Topsail. a Swell from the SW
Close Reef in the Topsails
H[ailed] a Ship
Moderate and fair Weather. the Swell from the SW Continuous.
HM sloop Resolution on Saturday, September 16th, 1776:
Mod[erate] breeze & fair W[ind]
New mastic the log line
Down Studdingsail & in 1st Reef Main & fore tops[ails]
Squally with showers. In 1st Reef M[i]z[zen]tops[ails]
D[itt]o W[ind]
Fresh Breezes & fair
Set the Foretopmast Studdingsail
Clean'd & smoak'd between Decks
Fresh Gales & fine W[inds] - Found the Ship 16 Miles to the No[rth]ward of her Rock
HM ship Director on Tuesday, September 24th, 1799:
Strong Breezes & Cloudy- A great Sea from the North which makes the Sails flap so violently that we dare not extend them.
Fresh Breezes & fair W[ind]
Saw this Man of War Birds & some Porpoises.
At 6 [PM]: The Sea a little less- out 2nd Reef of the Main S[ail]
Moderate & fair W[ind] - The Sea abating, out 2nd Reef Fore S[ail]
The Sea from the Northw[est] less, but still considerable.
At 5 [AM]: Moderate and Cloudy, fresh, veering at times.
Moderate & Fair W[inds]. Set the Fore Top Mast Stud[din]g Sails & Main Stud[din]g Sail
The fine W[ind] gives us opportunity to dry & air the Ship & all Clothes & Bedding
Unbent the small Bower Cable
Although there isn't much in there to show the author's inner thoughts or musings, I think they still do tell us a great deal about the human beings involved, and the originals especially have a vibrancy that is difficult to grasp in transcription. The little details like the loops of a given master's mate's handwriting, or a painstaking pencil sketch of the Cape of Good Hope, can really breathe life into the stories these logs hold. It's not quite the same as Captain Picard's thoughts from the bridge of the Enterprise, but there's still a rich humanity to be found in the humdrum records of what it took to sail these vessels.
I can offer some insight into a different time period and organisation from the interesting answer from u/fell-like-rain:
The Dutch East India Company (VOC) made it compulsory for captains to keep official logs. The example I have access to is from the early 17th century, when the Dutch, Portuguese and English were vying for control of the spice trade in Southeast Asia. Each organisation kept its own set of maps and navigational instructions, thus, information was extremely valuable. After every voyage, the logbook would be handed over to the VOC. The company cartographers would then use them to update their maps and charts with the latest information.
As with the logbooks mentioned by u/fell-like-rain there was a standard format to these logbooks. The left page was divided into 12 columns for the recording of date, course steers, distance sailed, estimated latitude and longitude, variation of the compass and the direction and force of the wind.
The right page contains the main findings and other thoughts, and it is here that we get closer to the kind of Captain's Log that we see in Star Trek with the recording of details of foreign cultures and strange new worlds. For example, in the 1640s, Captain John Anderson, a Scotsman in service of the VOC, wrote in his log that the inhabitants of Madagascar spoke 3 languages:
Malaya wich they as ye Hebrues from ye right hand to ye left, Syam from ye left hand to ye right as we, China right doune, all three wssfull and affected by the Industrous.
i.e. they write Malay from right to left, like Hebrew, they write Syam from left to right like us, and they write Chinese downwards.
Anderson's log contains many such observations that would seem out of place in a modern day log. It even contains an account of his participation in the VOC blockade of Portuguese controlled Goa. It reads more like a cross between true adventure and a travel guide than a log.
There were two reasons for including all these details. The first was that the Europeans were still gathering information about Asia, and thus the more details the better. Apart from information about weather and the like, captains often described the shape of landmasses, landmarks, customs, anything that might be useful to future captains sailing through or seeking to exploit trade opportunities.
The second was that this was an era when travel journals were popular reading. Willem Ysbrandtszoon Bontekoe, another captain of the VOC, published his travel journals in 1646, full of exotic locations and adventure, and these were very popular indeed. While the log of the above mentioned John Anderson was never published, the surviving copies we have include a paragraph at the end where he addresses readers directly, so it seems he was hoping to be published.
Captains with such aspirations would thus record as much information as possible in their logs. They would then make a copy and add on notes, perhaps about their background or the personalities of their crew. Then they would send them to a publisher, hoping that the publisher would take an interest, edit them for general consumption, and then print them for the mass market.
Despite these details, it does not seem common for VOC captains to overtly record their thoughts and feelings i.e. it was common to describe an interesting fruit's colour, shape and smell, perhaps, but not the way it made the captain feel. Even Anderson's descriptions of his participation in the Goa blockade are just descriptions of action, and lack details of whether he felt excited, frightened, whether he had any philosophical musings etc.
Enthoven, V., Murdoch, S., & Williamson, E. (2010). The Navigator. The Log of John Anderson, VOC Pilot Major, 1640-1643. (European Expansion and Indigenous Response; No. 7). Brill.