Baker was born in London. After serving an apprenticeship with a bookseller, he devised a system of instructing the deaf and dumb, by the practice of which he made a considerable fortune. This caught the attention of Daniel Defoe, whose youngest daughter Sophia he married in 1729.
A year before, under the name of Henry Stonecastle, he was associated with Defoe in starting the Universal Spectator and Weekly Journal. In 1740 he was elected fellow of the Society of Antiquaries and of the Royal Society. He contributed many memoirs to the Transactions of the Royal Society, and in 1744 received the Copley gold medal for microscopical observations on the crystallization of saline particles.
He was one of the founders of the Society of Arts in 1754, and for some time acted as its secretary. He died in London. Among his publications were The Microscope made Easy (1743), Employment for the Microscope (1753), and several volumes of verse, original and translated, including The Universe, a Poem intended to restrain the Pride of Man (1727). His name is perpetuated by the Bakerian lecture of the Royal Society, for the foundation of which he left by will the sum of £100.