Jónas was born in the north of Iceland, in Öxnadalur in Eyjafjörður. He was the son of Hallgrímur Þorsteinsson, a curate, and Rannveig Jónasdóttir. He was the third of their four children; his siblings were Þorsteinn (born 1800), Rannveig (born 1802) and Anna Margrét (born 1815). In 1816 Jónas' father drowned in a lake and Jónas was sent to live with his aunt. In 1821 he returned home to Öxnadalur to be confirmed, before going away to a school in Skagafjörður, where he was taught by the Reverend Einar H. Thorlacius. He studied there for two years, and won a scholarship to attend the school at Bessastaðir for a further six.