A type of hard candy, AKA a Jackson ball. These were a common type of penny candy in the early half of the 20th century. The flavoring would have been up to the candy maker, but the fact that they were referred to as Bull's Eyes rather than a more specific name indicates that they were fairly nondescript in flavor.
Nowadays you will often find soft candies with a cream or caramel center referred to as bull's eyes, and there is one soft caramel cream candy of the brand name Bulls-Eyes.
DonJaime reports that in the 1960s-70s in England Bull's Eyes were big round hard sweets which changed colour as you sucked them. Nowadays, in America, at least, these would be called jawbreakers or perhaps gobstoppers.