How to store more then max_size objects on a vector, which is size_t

/*
 * My code snippet (code.cpp)
 */
    for(item = 0; item < m_NumSetShipsAtt.size(); item++)
	{
        // create attacking objects; for every ship 1 object
		obj = FillObj(m_NumSetShipsAtt[item].Type, ATTER, m_NumSetShipsAtt[item].OwnerID);
        if(m_NumSetShipsAtt[item].OwnerID != CurrPl)
            CurrPl = m_NumSetShipsAtt[item].OwnerID;
 
        for(size_t o = 0; o < m_NumSetShipsAtt[item].Num; o++) {
        	try {
				m_AttObj->push_back(obj);
        	}
        	catch (bad_alloc& xa) {
 
        	    cerr << "In catch block, so an exception occurred: " << xa.what() << endl;
        	    cerr << "ships added so far: " << o << endl;
        	    cerr << "m_AttObj's size is: " << m_AttObj->size() << endl;
        	    cerr << "m_AttObj's capacity is: " << m_AttObj->capacity() << endl;
        	    cerr << "m_AttObj's max size is: " << m_AttObj->max_size() << endl << endl;
        	    cerr << "Exiting...";
        	    return 1;
        	}
        }
	}
 
/*
 * m_AttObj declaration snippet (code.h)
 */
	vector<Obj>* m_AttObj, *m_DefObj;
 
/*
 * vector declaration snippets (libstd++'s stl_vector.h)
 */
  template<typename _Tp, typename _Alloc = std::allocator<_Tp> >
    class vector : protected _Vector_base<_Tp, _Alloc>
    {
// ... much more
      /**  Returns the size() of the largest possible %vector.  */
      size_type
      max_size() const
      { return _M_get_Tp_allocator().max_size(); }
// ... much more again