Sveinn was born in Copenhagen, Denmark, as the son of Björn Jónsson (editor and later minister) and Elísabet Sveinsdóttir.

Although Iceland had become a sovereign state in 1918, its foreign affairs had been conducted by Denmark until the beginning of the Second World War.

In July 1941, United States troops occupied Iceland on the invitation of Sveinn's government and remained, in reduced numbers, after the war; their continued presence provoked the main controversy of the nation's postwar foreign policy.

Shortly after the creation of the republic, Sveinn called on Icelandic embassies to send him all books on diplomatic protocol that they could find, so that he could follow the right customs as head of state.

But, due to poor health, Sveinn was unable to visit Denmark during the final part of his tenure as president.

His eldest son Björn Sveinsson Björnsson served in the Nazi German military as a part of the Waffen-SS in World War II.

[10] His granddaughter, Brynhildur Georgía Björnsson, is the subject of the historical novel Konan við 1000° (The Woman at 1000°), written by Hallgrímur Helgason and published in 2011.

His great grandson is Henrik Björnsson, singer and lead guitarist in the shoegaze rock band Singapore Sling.

[11] His great granddaughter is Georgía Olga Kristiansen, the first female referee to officiate in the highest competitive tier men's basketball league in Iceland.