There are literally thousands of statistical studies on this subject and they all pretty much agree:

1. To maintain a high level of genetic diversity requires at least 1000 individuals.
2. To guard against common environmental effects requires 1500 individuals.
3. A population of 10,000 individuals would have a 99% probability of not going extinct within the next 1000 years even with extreme environmental catastrophes (of the natural sort).

References:
MVP Under the Endangered Species Act

The living quarters for 10,000 people could easily fit inside a 500-meter diameter sphere. Under 1-g rotation each person could enjoy a 1000 square-foot bungalow-style home with 10-foot ceilings, and still take up less than 1.2 percent of the Habitable Volume. Of course there must be larger areas for hydroponics, recreation, industrial machines, etc.

With a 242,000 tonne weapons-grade uranium core to keep the temperature steady at 20 degrees C for the entire 860-year trip to Alpha Centauri I see no reason to expect environmental catastrophe - except those that would kill everyone on board instantly.