std::string is perfectly capable of storing nulls. However, you have to be wary, as const char* is not, and you very briefly construct a const char*, from which you create the std::string.
std::string a("\x00");
This creates a constant C string containing only the null character, followed by a null terminator. But C strings don't know how long they are; so the string thinks it runs until the first null terminator, which is the first character. Hence, a zero-length string is created.
std::string b("");
b.push_back('\0');
std::string is null-clean. Characters (\0) can be the zero byte freely as well. So, here, there is nothing stopping us from correctly reading the data structure. The length of b will be 1.
In general, you need to avoid constructing C strings containing null characters. If you read the input from a file directly into std::string or make sure to push the characters one at a time, you can get the result you want. If you really need a constant string with null characters, consider using some other sentinel character instead of \0 and then (if you really need it) replace those characters with '\0' after loading into std::string.