The size of a string is only limited by the amount of memory available to the program, it is more of a operating system limitation than a C++ limitation. C++/C strings are null terminated so the string routines will happily process extremely long strings until they find a null.
On Win32 the maximum amount of memory available for data is normally around 2 Gigs.
You can read arbitrarily large amounts of data from a socket, but you must have some way of delimiting the data that you're reading. There must be an end of record marker or length associated with the records that you are reading so use that to parse the records. Do you really want read the data into a string? What happens if your don't have enough free memory to hold the data in RAM? I suspect there is a more efficient way to handle this data, but I don't know enough about the problem.