Like WildDesu and CarrotLord, I don't play a whole lot of modern games, unless you count random internet flash games. Other than my computer, which is itself about 6 years old and is therefore a bit limited in what games it can run, the newest console system that I ever had growing up was the SNES. My brother and I thought about getting an N64 or Gamecube back in the day, but somehow never got around to it. Actually, that's not entirely true about the SNES; my mom bought a PS2 while I was in high school or something for the sole purpose of playing DDR. Too bad we never got a memory card to go with it.
Oh, and the newest handheld system that I have is a GBA. So yeah, I'm something of a "games of ages past" person, I guess.
Anyway, regarding what makes a game good, for me it has to do with a balance of all the factors listed in the poll, and maybe some others, too. Also, I don't think there's one secret formula for every game. Some games are designed to be more story-driven while others rely less on story but more on gameplay. A game can be great either way, but the purpose of why you play a game is an important factor, too. If all you want is some mindless fun, you probably are not going to want something with deep, heavy story like a Final Fantasy game or any RPG at all for that matter. Something like an old Rampage game or a space shooter or bullet hell game might be more enjoyable. A lot of arcade games are classics (maybe even "great games?" Think PacMan or Asteroids), but don't have a whole lot in the way of storyline.
If instead you want to immerse yourself in another world where you meet interesting characters and engage in a more cerebral adventure, something that's heavily story-based is probably going to be more appealing than a shoot-em-up action game. Puzzle games tickle another part of the mind during a gaming experience, and of course every different kind of game has its own appeal for its own reasons. What makes a game good is that it fills a particular niche of enjoyment by being balanced in different ways.
An aspect of gaming that I mentioned in irc a few days ago was a game's script. By that I mean the dialogue or narration flows, how events unfold (especially in cut scenes), and things like that. We decided that it fit pretty well under the category of "story," but in my mind there is still a distinction between the two. A game can have a good plot or premise, that is a good story, although the dialogue or cut scenes can be a bit strained. Especially with dialogue, sometimes the words that come on the screen just don't quite work but you know what they're trying to say. In this case, the story could still be good while the writing is lacking. Maybe this has to do with a funky translation or the script writers just had a bad day, but a wonky script can have either a diminishing or endearing effect on a game apart from the overall quality of its story.
tl;dr: Different games thrive on various combinations of and emphases on all the sundry aspects of a game.
When a game fails, whether modern or not, it is often because it is unbalanced: it relies too heavily on one or some elements (especially graphics, today) at the expense of others.