Does the game eventually stop? Or can I just keep on playing. I know you have the option to "Keep playing a few more turns" even after the victory conditions are met, but I've heard that the game always stops at the year 2250 or something like that.
No. The scoring ends at 2050, so if you didn't achieve any of the 5 official kind of victories by then, you'll win by "Rank" if you are first in the Historigraph scoring. Believe me, if you let enabled the Space Race victory, you or the AI will get it much before depending of the difficulty.
__________________ The art of mastering:"la Maîtrise des caprices du subconscient avant tout".