Civilization's fun is unfortunately coupled with lengthy repetition
The unique challenges Civilization displays, are in my opinion, where it shines. Countless times I have attacked an enemy city, only to realize their defense was much more powerful then anything I could attack it with. So I think logically. I either gather huge amounts of my forces to attack it at once, or maybe I just have to wait until I discover how to build more powerful units. This type of war scenerio is always fun, because you realize how you're thinking, whether you catch yourself building a city near their city and making the units there, or calling for a peace treaty to buy yourself some time, you realize you're thinking strategically, and it's fun.
That is, until the end of the game. If all is going well, you should be crushing your enemies and the only thing you have to do every turn is build some things you don't need, and destroy enemies who pose no threat. It becomes almost a chore, and this sort of monotony is what really damages the game.
As for multiplayer, it's functional, but not very enjoyable. It's an all out war from the start, but none of you have time to prepare and make things interesting. The diplomacy panel is useless, and there's rarely time for much strategy of any sort.
In summary, Civilization does have some very irritating flaws, but is certainly worth playing.