Class variables: public access read-only, but private access read/write

Of course you can:

class MyClass
{
    int x_;

public:
    int x() const { return x_; }
};

If you don't want to make a copy (for integers, there is no overhead), do the following:

class MyClass
{
    std::vector<double> v_;

public:
    decltype(v)& v() const { return v_; }
};

or with C++98:

class MyClass
{
    std::vector<double> v_;

public:
    const std::vector<double>& v() const { return v_; }
};

This does not make any copy. It returns a reference to const.

While I think a getter function that returns const T& is the better solution, you can have almost precisely the syntax you asked for:

class myClass {
    private:
    int x_; // Note: different name than public, read-only interface

    public:
    void f() {
        x_ = 10; // Note use of private var
    }
    const int& x;
    myClass() : x_(42), x(x_) {} // must have constructor to initialize reference
};

int main() {
    myClass temp;

    // temp.x is const, so ...
    cout << temp.x << endl; // works
    // temp.x = 57;  // fails

}

EDIT: With a proxy class, you can get precisely the syntax you asked for:

class myClass {
public:

    template <class T>
    class proxy {
        friend class myClass;
    private:
        T data;
        T operator=(const T& arg) { data = arg; return data; }
    public:
        operator const T&() const { return data; }
    };

    proxy<int> x;
    // proxy<std::vector<double> > y;


    public:
    void f() {
        x = 10; // Note use of private var
    }
};

temp.x appears to be a read-write int in the class, but a read-only int in main.