SHANGHAI—China’s coronavirus epidemic is depressing its economic outlook, with new government readings on the manufacturing and service sectors validating informal indications that the country is struggling to get back to work.

A Chinese government index that tracks sentiment among purchasing managers at manufacturers fell to its lowest level on record in February, dropping deep into territory that indicates a contraction. China’s National Bureau of Statistics said Saturday that its 15-year-old index tumbled to 35.7 from...