Lots of the 2600 library is still eminently playable today. Games don't have to be "modernish" to hold up well. A good game is a good game from any period. Someone mentioned chess. Sure not a video game but the thread title didn't specify video games so technically an excellent answer to the question.
I said "modernness."
To an eye looking back, some of these games can feel very modern in setting, style, and set-up. For instance, one of the DnD games on the Intellivision is a first-person dungeon crawler with surprisingly modern elements, all from an era where 3-D movement didn't really exist.
The most important part, I think, is control. If the game controls in a way that a modern gamer will instantly recognize and be able to adapt to, it feels as though it could be a modern game--that's timeless, that "ages" well because, in a large part, it may get older, but it doesn't actually age.
I mentioned Galaga because it still feels fresh and modern--shmups prior to Galaga feel dated and archaic. Space Invaders is slow and clunky. Galaxian is just slow. But Galaga nailed the speed of attacks, the intensity of the gameplay, and the smooth, solid controls.
I don't think its the modern feel. I just think fun is fun is fun. If you have fun with a game, it's a good game. If a game is amazing because of some technical advancement, the fun may not be there when the shiny new wears off.
That's just my 2 cents. That's why Galaga, Pac-Man, Super Mario Bros. are all still fun.
I say "modern" referring to how game design evolves. Some things are no longer really acceptable--like random deaths, or trial-and-error gaming, or chunky controls. "Modern" game design requires a certain level of smoothness and precision in general game controls, modern game design tends to educate the player without resorting to bad trial-and-error gameplay concepts, and the like.
Granted, there are a lot of modern games that fail at this, and they tend to be viewed thusly.
Galaga's gameplay largely still feels modern, or could work in a new, contemporary game.
Galaga feels like any other modern, well-respected, easily playable shmup. It's play, it's flow, it's control. Essentially, Galaga
was the first modern shmup, in my opinion. That's why it still holds up today. Control-wise, having gimmicks, and a weapon upgrade--these things add to it. Granted, weapon upgrades in shmups have grown and changed and the like through the years, so the
way Galaga does it's upgrade is archaic, but that it feels like an Easter Egg probably helps it.
I agree that a fun game is still fun, but I'm going into the
why of that, because that is extremely interesting to me. It's something I've thought about a lot, analyzed a lot. Why is such-and-such game still so relevant, still so fresh and/or playable, while such-and-such from fifteen years later feels so old and dated?
A great way to see examine this is to compare Galaga to Galaxian. Galaxian has aged horribly, and feels like a bizarre stop-gap between Space Invaders and Galaga, which it is. I could, for example (personally) go from Ikaruga to Galaga, and be able to transition well and find the experience still totally relevant. Going from Ikaruga to Galaxian is not so smooth. It's aged poorly.