Doom historically has been ported to everything from the Super Nintendo to pregnancy tests and now the new Nintendo Game & Watch. The new Game & Watch isn't very powerful and it looks like quite a bit was required to get it to run at all. The system isn't as easy to hack as the NES or SNES classic but if you have some time on your hands it looks like Doom will run but doesn't play very well as of yet.

The Game & Watch is also quite limited in terms of storage (1.1 Mb) so even if you do hack it doesn't seem like it would be able to store an entire NES ROM set.