Is it possible to make a field of a record private? or to make a member of record private?

Nope, it's not possible for single fields to be private: http://msdn.microsoft.com/en-us/library/dd233184

However, you can make all fields private and expose selected fields through properties. Note that you will need a Create-function in order to create an instance of the records, since its' fields are private:

type MyRec = 
    private
        { a : int
          b : int }
    member x.A = x.a
    member private x.Both = x.a + x.b  // Members can be private
    static member CreateMyRec(a, b) = { a = a; b = b }

Members can be private though, as in the case of the MyRec.Both property above.

Edit: The above makes the fields private to the module that MyRec is defined in, not private to MyRec. See Daniel's answer to Weird accessibility scopes when F# record's fields are declared private.