std::vector<std::vector<int>> main_vector;
std::vector<int> temp_vector;

for (std::size_t i = 0; i < 5; i++)
{
    main_vector.push_back(temp_vector);
    for (std::size_t j = 0; j < 5; j++)
    {
        main_vector[i][j] = j;
    }
}

bin_file.open("test.txt", std::ios::out | std::ios::binary);
bin_file.write((char *)&main_vector, sizeof(main_vector)); //не сработает