I got some premade PCBs on which I soldered ROM chips containing my game. Wrapped this in a NES cartridge shell. Finally, I printed stickers and a manual.
The exact process is pretty long to explain. While fascinating to do, it will not fit in a comment. You can find more by searching for “NES repro tutorial” on the internet. A “repro” the re-creation of an old game, but it is exactly the same process for your own new game (and a lot more legal.)