Scott was born on 6 June 1868, the third of six children and elder son of John Edward, a brewer and magistrate, and Hannah (née Cuming) Scott of Stoke Damerel, near Devonport. There were also naval and military traditions in the family, Scott's grandfather and four uncles all having served in the army or navy. John Scott's prosperity came from the ownership of a small Plymouth brewery which he had inherited from his father and subsequently sold. Scott's early childhood years were spent in comfort, but some years later, when he was establishing his naval career, the family would suffer serious financial misfortune.
In accordance with the family's tradition, Scott and his younger brother Archie were predestined for careers in the armed services. Scott spent four years at a local day school before being sent to Stubbington House School in Hampshire, a cramming establishment that prepared candidates for the entrance examinations to the naval training ship HMS Britannia at Dartmouth. Having passed these exams Scott began his naval career in 1881, as a 13-year-old cadet.
In July 1883, Scott passed out of Britannia as a midshipman, seventh overall in a class of 26. By October, he was en route to South Africa to join HMS Boadicea, the flagship of the Cape squadron, the first of several ships on which he served during his midshipman years. While stationed in St Kitts, West Indies, on HMS Rover, he had his first encounter with Clements Markham, then Secretary of the Royal Geographical Society, who would loom large in Scott's later career. On this occasion, 1 March 1887, Markham observed Midshipman Scott's cutter winning that morning's race across the bay. Markham's habit was to "collect" likely young naval officers with a view to their undertaking polar exploration work in the future. He was impressed by Scott's intelligence, enthusiasm and charm, and the 18-year-old midshipman was duly noted.
In March 1888 Scott passed his examinations for sub-lieutenant, with four first class certificates out of five. His career progressed smoothly, with service on various ships and promotion to lieutenant in 1889. In 1891, after a long spell in foreign waters, he applied for the two-year torpedo training course on HMS Vernon, an important career step. He graduated with first class certificates in both the theory and practical examinations. A small blot occurred in the summer of 1893 when, while commanding a torpedo boat, Scott ran it aground, a mishap which earned him a mild rebuke.
During the research for his dual biography of Scott and Roald Amundsen, polar historian Roland Huntford investigated a possible scandal in Scott's early naval career, related to the period 1889–90 when Scott was a lieutenant on HMS Amphion. According to Huntford, Scott "disappears from naval records" for eight months, from mid-August 1889 until 26 March 1890. Huntford hints at involvement with a married American woman, a cover-up, and protection by senior officers. Biographer David Crane reduces the missing period to eleven weeks, but is unable to clarify further. He rejects the notion of protection by senior officers on the grounds that Scott was not important or well-connected enough to warrant this. Documents that may have offered explanations are missing from Admiralty records.
In 1894, while serving as torpedo officer on the depot ship HMS Vulcan, Scott learned of the financial calamity that had overtaken his family. John Scott, having sold the brewery and invested the proceeds unwisely, had lost all his capital and was now virtually bankrupt. At the age of 63, and in poor health, he was forced to take a job as a brewery manager and move his family to Shepton Mallet, Somerset. Three years later, while Robert was serving with the Channel squadron flagship HMS Majestic, John Scott died of heart disease, creating a fresh family crisis. Hannah Scott and her two unmarried daughters now relied entirely on the service pay of Scott and the salary of younger brother Archie, who had left the army for a higher-paid post in the colonial service. Archie's own death in the autumn of 1898, after contracting typhoid fever, meant that the whole financial responsibility for the family rested on Scott.
Promotion, and the extra income this would bring, now became a matter of considerable concern to Scott. In the Royal Navy however, opportunities for career advancement were both limited and keenly sought after by ambitious officers. Early in June 1899, while home on leave, he had a chance encounter in a London street with Clements Markham, who was now knighted and President of the Royal Geographical Society (RGS), and learned for the first time of an impending Antarctic expedition with , under the auspices of the RGS. It was the opportunity for early command and a chance to distinguish himself, rather than any predilection for polar exploration which motivated Scott, according to Crane. What passed between them on this occasion is not recorded, but a few days later, on 11 June, Scott appeared at the Markham residence and volunteered to lead the expedition.