Is it possible to program in binary?

Of course. It’s more commonly called machine code. It’s basically assembly language without the mnemonic devices. Someone who knows assembly very well could program in machine code with additional effort, referring to opcode listings (e.g. x86) as needed.

Would I do it? No. Even assembly is only useful in rare circumstances, and there’s no reason (beside demonstrating your skills) to reject the assembler‘s help.

Since you asked about hello world, you should check out this article. He shows how he wrote, then optimized, an x86 ELF program to output it. It was originally written in nasm then modified in a hex editor.

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