Born in Adelaide, South Australia, Barwell was educated at St Peter's College and Adelaide University, graduating in law. Admitted to the bar in 1899, Barwell built a successful legal practice where he specialised in defending murder suspects and became a prominent figure in the Adelaide Establishment. In 1902, he married Anne Webb in Clare, South Australia and together they had one son and three daughters.
Barwell entered the South Australian House of Assembly in 1915 as the Liberal Union member for the seat of Stanley. In parliament he quickly became known both as an uncompromising conservative and as a likely future premier. He defended the restricted franchise of the South Australian Legislative Council, arguing that the Australian Labor Party should not be allowed to gain control "over the capital that employs labor, and over the superior intellect that governs that labor".
In 1917, Barwell was made Attorney-General of South Australia and Minister for Industry in Archibald Peake's cabinet and was forced to deal with the deteriorating relationship between the urban and rural constituencies of the Liberal Union, which worsened with the creation of the Country Party in 1919, taking many of the Liberal Union's supporters with it. Peake died soon after and Barwell became Premier of South Australia on 8 April 1920. Despite voter antipathy against Barwell over his abrasive and sometimes tactless political style, the Liberal Union was nonetheless returned to office at the 1921 election, with Barwell retaining the Premiership.
Never one to shy away from controversy, Barwell publicly advocated the importation of coloured labour into tropical Australia, contrary to the White Australia Policy which at the time had almost unanimous support. Barwell argued that northern Australia had proven unsuitable for white settlement and only the large scale importation of "selected Asiatics working as coolie labour under indenture to white men" would help develop the region as they were the only race suited to such conditions. This was heresy for many Australians, particularly those in the opposition Labor Party, for which the White Australia Policy was historically a central plank in their policy platform. Labor politicians treated Barwell with undisguised contempt for the rest of his career, referring to him as "Black Barwell".
Recognising the need to improve South Australia's railways, Barwell authorised significant expenditure on laying new tracks, the purchase of new trains (which became known as "Barwell Bulls") and on the construction of the grand Adelaide railway station. Unfortunately, the rising popularity of the automobile left many of Barwell's rail projects obsolete as soon as they were completed.
In 1922 Barwell announced the "South Australian Farm Apprenticeship Scheme", which undertook the ambitious target of arranging the immigration of 6 000 young men and boys from England to cover the 6 000 South Australian World War I casualties. He travelled to England to personally oversee the recruitment of what became known as the "Barwell Boys". Widespread unemployment in England led 14 000 boys, mainly aged between 15 and 17, to apply for the scheme. While the eventual number of Barwell Boys numbered only 1 700, the scheme was considered a great success, with many of the émigrés playing significant roles in the development of South Australia.
While in London, Barwell was made a Knight Commander of the Order of St Michael and St George (KCMG) in the Birthday Honours List of 1922. He returned to South Australia to find that he and his government were becoming increasingly unpopular due to his policies of small government and wage restraint, as well as lingering resentment over his earlier comments on importing coloured labour.
Barwell lost the 1924 election to the John Gunn led Labor Party and, after briefly acting as Opposition Leader, resigned from state parliament, seeking a seat in the Australian House of Representatives with a view to becoming Prime Minister. No seats were immediately forthcoming and instead he was forced to settle for appointment to the Senate, filling a vacancy caused by the death of Senator James O'Loghlin in 1925.
Sitting with the Nationalist Party of Australia, Barwell served in the Senate until 1928, often clashing with his party colleagues due to his outspokenness and independent mind. Realising that a move into the lower house was now a forlorn hope, Barwell resigned from the Senate to accept the posting of South Australian Agent-General in London. He served in that position until 1933, helping to prepare opinion for the Ottawa Agreement and for the closer collaboration of the various parts of the British Empire. Described by The Times as a "strong Imperialist with a practical outlook", Barwell was a firm believer in reciprocal trade between members of the Empire.
After the completion of his term as Agent General, Barwell remained in London, entering into various business interests, before eventually returning to Adelaide in 1940, where he unsuccessfully stood for pre-selection in his old seat of Stanley. Growing increasingly deaf, Barwell served as Deputy Chairman of the South Australian Housing Trust for fifteen years until his death in 1959 from cerebrovascular disease.