... can someone gather/hunt/farm all of the nutrients required to be healthy whilst never having to land again?
Nutrients yes, healthy not likely.
A 1000 days is a grueling journey.
Reid Stowe, whom used to have a website called (Beyond) 1000 Days at Sea: The Mars Ocean Odyssey, lived at sea (without contact with land) for 1,152 days (equals 3 Years, 1 Month, 3 Weeks, 5 Days, 13 Hours, 26 Minutes and 39 Seconds).
Stowe is an experienced mariner and was the principal designer and builder of the Anne, a 70 ft (21.3 m), 60-ton (54,400 kg) gaff-rigged schooner which he sailed on this voyage. The purpose of the enterprise was to remain on the open ocean, without resupply or pulling into any harbor, for a period of one thousand days.
He took a lot of supplies with him and grew simple plants (like beanspouts) while at sea.
The heavily laden schooner passed through New York Harbor and into the open ocean by the evening of April 21 2007. ... On Day 658, Reid Stowe broke the world record for the longest non-stop ocean voyage, previously held by Jon Sanders, if one disregards Nansen's Fram expedition, during which the schooner Fram lay trapped on ice for nearly three years, and the crew was away from land for at least 1067 days.
Reid Stowe and his support team have since accomplished one of their goals of a person sailing on the open seas without resupply for 1000 days, as well as breaking the 1067-day record set by the Fram in 1896. January 16, 2010 was officially the day of the 1000-day mark, while March 24, 2010 equalled the 1067-day mark. Subsequent to the first 306 days with Soanya Ahmad, Reid Stowe also broke the record for the longest solo sea voyage without resupply, on Day 964 (Dec. 11, 2009). Furthermore, as a two-member male-and-female crew, Stowe and Ahmad could also lay claim to the longest non-stop voyage on the ocean by a man and a woman since Bernard Moitessier and his wife Françoise completed a 126-day voyage in 1966, from Tahiti to Spain.
None of the records claimed by Stowe were ever officially recognized. The World Sailing Speed Record Council explains on their website: "We concentrate on speed record attempts and claims, and no longer recognize "human condition" categories which can expand to such an extent that almost anyone would be able to claim a record of some sort."
Throughout the journey, Stowe maintained contact with the New York City–based support team via an Iridium phone. Stowe employed a VHF marine transceiver for ship-to-ship communications. Volunteers maintained a web site so that the general public could follow the progress of the voyage. The entire route of the schooner Anne was verified daily by GPS tracking, and the manufacturer of the equipment has made the database available online.
Guinness World Records:
13 March 1988
The longest distance sailed non-stop by any vessel is 71,023 nautical miles (131,535 km; 81,732 miles), a feat achieved by Australian Jon Sanders between 25 May 1986 and 13 March 1988. Starting and finishing in Fremantle, Western Australia, Jon made a record three consecutive non-stop solo circumnavigations of the globe – one west and two east – and was at sea for a total of 657 days on his 13.9-m (44-ft) sloop Parry Endeavour.
No doubt you could load a cargo vessel with a lifetime worth of supplies but there are practicalities such as maintenance of the vessel and your own health.
Pescetarianism or pescatarianism (/ˌpɛskəˈtɛəriənɪzəm/) is the practice of adhering to a diet that incorporates seafood as the only source of meat in an otherwise vegetarian diet. Most pescetarians are ovo-lacto vegetarians who eat seafood along with dairy products and eggs, often colloquially defined as "fish but no other meat".
Yes, it is considered a healthy diet.
If you can make your own repairs at sea, as Stowe did, and never need to visit a doctor (or a hospital) then you can live at sea indefinitely.