How to write and execute PURE machine code manually without containers like EXE or ELF?

Everyone knows that the application we usually wrote is run on the operating system. And managed by it.

It means that the operating system is run on the machine. So I think that is PURE machine code which you said.

So, you need to study how an operating system works.

Here is some NASM assembly code for a boot sector which can print "Hello world" in PURE.

 org
   xor ax, ax
   mov ds, ax
   mov si, msg
boot_loop:lodsb
   or al, al 
   jz go_flag   
   mov ah, 0x0E
   int 0x10
   jmp boot_loop

go_flag:
   jmp go_flag

msg   db 'hello world', 13, 10, 0

   times 510-($-$$) db 0
   db 0x55
   db 0xAA

And you can find more resources here: http://wiki.osdev.org/Main_Page.

END.

If you had installed nasm and had a floppy, You can

nasm boot.asm -f bin -o boot.bin
dd if=boot.bin of=/dev/fd0

Then, you can boot from this floppy and you will see the message. (NOTE: you should make the first boot of your computer the floppy.)

In fact, I suggest you run that code in full virtual machine, like: bochs, virtualbox etc. Because it is hard to find a machines with a floppy.

So, the steps are First, you should need to install a full virtual machine. Second, create a visual floppy by commend: bximage Third, write bin file to that visual floppy. Last, start your visual machine from that visual floppy.

NOTE: In https://wiki.osdev.org , there are some basic information about that topic.