The kind of Eruv that we're talking about (with wire, rather than an actual wall) only applies to a karmelit, an area which is neither public nor private by the

The kind of Eruv that we're talking about (with wire, rather than an actual wall) only applies to a karmelit, an area which is neither public nor private by the biblical definition (or how the rabbis read the biblical definition). Carrying in a karmelit was not prohibited by the Torah itself, but was prohibited by the rabbis to prevent people from accidentally violating the biblical prohibition. In this case, they're relaxing a rule that they themselves created. In fact, some of the controversy around building Eruvin in large cities like Manhattan is that large cities may have certain features that make them a public domain according to the biblical prohibition, thus making an Eruv of this kind impossible.