It's usually good to add streaming operators to the class/struct since it breaks down a big problem into smaller parts. I've made an example here. I've added a class (OrderBox) for keeping many Order objects and added streaming operators for that class too. I don't write the length of the FIO field to the file but instead read/write until \0. Since \0 is perfectly valid in the middle of a string (but I assume that you will not use strings like that), I make sure that \0 in the middle of a string is never saved to file (look for the std::strlen in Orders ofstream operator). Should work in C++14/17/2a. 
#include <iostream>
#include <initializer_list>
#include <fstream>
#include <vector>
#include <cstring>
#include <stdexcept>
enum Products {OOPlabs}; // no idea if this is an enum, needed it to compile ...
//-----------------------------------------------------------------------------
// slightly restructured - I put the FIO field last
struct Order
{
    unsigned int productamount = 0;
    Products product = Products::OOPlabs;
    double cost = 0.0;
    unsigned int orderID = 0;
    std::string FIO = "Иванов Иван Иванович";
    // operator to read from file
    friend std::ifstream& operator>>(std::ifstream&, Order&);
    // operator to write to file
    friend std::ofstream& operator<<(std::ofstream&, const Order&);
    // operator to stream to other ostreams, like std::cout
    friend std::ostream& operator<<(std::ostream&, const Order&);
};
using OrderVec = std::vector<Order>;
//-----------------------------------------------------------------------------
// read one Order from a file stream
std::ifstream& operator>>(std::ifstream& is, Order& ord) {
    is.read(reinterpret_cast<char*>(&ord.productamount), sizeof(ord.productamount));
    is.read(reinterpret_cast<char*>(&ord.product), sizeof(ord.product));
    is.read(reinterpret_cast<char*>(&ord.cost), sizeof(ord.cost));
    is.read(reinterpret_cast<char*>(&ord.orderID), sizeof(ord.orderID));
    std::getline(is, ord.FIO, '\0');
    return is;
}
// write one Order to a file stream
std::ofstream& operator<<(std::ofstream& os, const Order& ord) {
    os.write(reinterpret_cast<const char*>(&ord.productamount), sizeof(ord.productamount));
    os.write(reinterpret_cast<const char*>(&ord.product), sizeof(ord.product));
    os.write(reinterpret_cast<const char*>(&ord.cost), sizeof(ord.cost));
    os.write(reinterpret_cast<const char*>(&ord.orderID), sizeof(ord.orderID));
    // using strlen in case a '\0' has snuck into the string
    os.write(ord.FIO.c_str(), std::strlen(ord.FIO.c_str())+1);
    return os;
}
// stream an Order ... to std::cout
std::ostream& operator<<(std::ostream& os, const Order& ord) {
    os << "{" << ord.productamount << "," << ord.product << "," << ord.cost
       << "," << ord.orderID << "," << ord.FIO << "}";
    return os;
}
//-----------------------------------------------------------------------------
class OrderBox { // to keep all your Order objects
    OrderVec m_orders;
public:
    // default ctor
    OrderBox() : m_orders() {}
    // ctor to read from a file
    OrderBox(const std::string file) :
        m_orders()
    {
        if(!readbinfile(file))
            throw std::runtime_error("error reading file");
    }
    // ctor to populate from an initializer_list {...}
    OrderBox(std::initializer_list<Order> il) :
        m_orders(il)
    {}
    bool createbinfile(const std::string& filename)
    {
        std::ofstream f(filename, std::ios::trunc | std::ios::binary);
        if(f) f << *this; // use OrderBox's ofstream operator
        return f.good();
    }
    bool readbinfile(const std::string& filename)
    {
        std::ifstream f(filename, std::ios::binary);
        if(f) f >> *this; // use OrderBox's ifstream operator
        return f.good();
    }
    // the OrderBox's stream operators
    friend std::ifstream& operator>>(std::ifstream&, OrderBox&);
    friend std::ofstream& operator<<(std::ofstream&, const OrderBox&);
    friend std::ostream& operator<<(std::ostream&, const OrderBox&);
};
//-----------------------------------------------------------------------------
std::ifstream& operator>>(std::ifstream& is, OrderBox& ob) {
    OrderVec result;
    Order tmpord;
    size_t reqlen;
    is.read(reinterpret_cast<char*>(&reqlen), sizeof(reqlen));
    result.reserve(reqlen);
    while(is>>tmpord) // use the ifstream operator of Order
    {
        result.emplace_back(std::move(tmpord));
        if(result.size()==reqlen) break; // all records read
    }
    if(result.size()!=reqlen) is.setstate(std::ios_base::failbit);
    else std::swap(result, ob.m_orders);
    return is;
}
std::ofstream& operator<<(std::ofstream& os, const OrderBox& ob) {
    size_t reqlen = ob.m_orders.size();
    os.write(reinterpret_cast<const char*>(&reqlen), sizeof(reqlen));
    for(const auto& ord : ob.m_orders)
    {
        os << ord; // use the ofstream operator of Order
    }
    return os;
}
std::ostream& operator<<(std::ostream& os, const OrderBox& ob) {
    os << "{\n";
    for(const auto& ord : ob.m_orders) {
        std::cout << " " << ord << "\n"; // print one order
    }
    os << "}\n";
    return os;
}
//-----------------------------------------------------------------------------
int main() {
    OrderBox ob = { // OrderBox with orders to write to file
        {10, OOPlabs, 0.1, 1, "Hello One"},
        {20, OOPlabs, 0.2, 2, "Hello Two"},
        {30, OOPlabs, 0.3, 3, "Hello Three"},
        {40, OOPlabs, 0.4, 4, "Hello Four"},
        {50, OOPlabs, 0.5, 5, "Hello Five"}
    };
    try {
        if(ob.createbinfile("orders.db")) {
            // a new OrderBox to populate directly from the file
            OrderBox newbox("orders.db");
            // stream OrderBox
            std::cout << newbox;
        }
    } catch(const std::exception& ex) {
        std::clog << "Exception: " << ex.what() << "\n";
    }
}
Output if it all works:
{
 {10,0,0.1,1,Hello One}
 {20,0,0.2,2,Hello Two}
 {30,0,0.3,3,Hello Three}
 {40,0,0.4,4,Hello Four}
 {50,0,0.5,5,Hello Five}
}