Depends on what you mean by "all the elements" of a story-based game, as you can totally
make something functional. However, it's not really any different than just making a multiplayer map thats only built for one person. If you can't make it happen in a multiplayer match, you can't make it happen here either.
(This is not completely accurate, as some entities will try to target "the player" and I believe only having one human will allow these entities to work in ways they might not in actual multiplayer servers, but those entities are few and far between. It's not something I have a lot of experience with.)