Well... besides the earth being flat and the literal glass dome preventing anyone from escaping... (jkjkjk)
Given enough time, anything is possible. But if we wanted to do this quickly, we would run into some serious road-blocks.
Finding a planet which is Earthlike enough
First of all, we need to find a planet similar to earth. Colonizing Mars might seem like a good idea, but its mass, nearness to the sun, lack of a moon like ours, and the many other differences between Mars and earth, make it likely that plant life won't grow there easily, and long-term exposure to that environment may have significant negative effects on other organisms. There are hundreds of factors involved in enabling life on earth: mass, density, temperature, atmosphere, nearness to sun, type of sun, mass/density of moon, nearness to moon, magnetic field, mineral contents of ocean, ordering of layers in crust, volcanic activity, shape of mountains affecting wind, etc. etc. etc.. For example, we know that long-term exposure to non-earthlike gravity can cause people's bones to atrophy. Every one of those conditions, if not met well, can lead to other serious health conditions. Given the time it will take to get there, we should get this right the first time, so take a few years thinking about where to go, and maybe send out a few probes.
There have been some theoretical ideas thrown around for how we can get a probe to fly-by a far planet which we think is earthlike in as little as 60 years using massive solar sails, but the probe was tiny, and we still had a problem where it couldn't decelerate quickly enough to orbit. At best, we would get a few flyby pictures at extremely fast speeds. Stopping in a foreign solar system within one lifetime is a problem we haven't figured out how to solve (and might not be able to solve because of the limits on acceleration which humans can tolerate). So, we'll have to first make a space station which can support plant and human life for several generations, which might be impossible even with spinning stations to simulate gravity, because of the inherent differences those stations will have when compared to earth.
(120 years or more)
Generation Ship Planning/Construction
Due to the careful planning of the ship-side ecosystem, population control on the generation ship will have to be strictly regimented. This will represent significant social engineering problems which humanity has never faced before. The few experiments with this kind of small isolated population, which I'm aware of, have been steaming failures. We really don't know how humans will handle spending their entire lives aboard a space ship with only a handful of types of food, their romantic interests potentially having been arranged several generations prior to maximize diversity, and the possibility that a tiny, simple mistake made by some kid on the ship could cause a breach or something and kill everyone. Not to mention, you'll have to have warehouses with new electronic components to constantly maintain your ship in-flight. Electronics don't last all that long.
(For this, assuming just one minor failure unaccounted-for, I'm adding an extra 20 years. But realistically, we should expect several failures)
Planetside Resources / Terraforming
If we can land the generation ship on the planet and call that acceptable, then skip this part -- but I think a "planet colony" should actually be interacting with the planet somehow, and not just making physical contact with it via ship struts.
Then, once we get there, we'll have immediate resource issues planetside. Earth represents a very distinct mix of materials favorable to life, and we might not be able to survive off-world without either frequently returning for supplies, or quickly gathering resources from several off-world planets simultaneously. All human colonies depend on transformation of one type of matter into another type of matter in order to produce energy, food, etc. Our plants depend on water with very specific ratios of specific minerals in it to bear fruit, and won't live with too much of this or that other chemical in the water.
Suppose there is too much of Chemical A in the water, so we harvest Chemical B to make filters. When it's time to recycle the filter, we can't recover all the B from the dirty filter, so we have to continually harvest more. Suppose B is not abundant in the soil, so we have to dig deep to find stores of it. To make matters worse, the outer crust of this planet is also dense with Chemical C, which is toxic to humans on contact, but the process of removing that requires lots of time, energy, and maybe even some Chemical D, which just isn't present on this planet at all. So we harvest D from a nearby moon or asteroid, use it to terraform a small portion of the surface, and set up deep and very dangerous mines in the planet to acquire B. In the meantime none of our plants will grow, so now we have to land our generation ship safely on the planet, or be constantly shipping supplies back and forth in orbit. Anyway, all that to say, Tweaking the planet to enable it to support life will take several more generations.
(At least another 100-200 years before a small part of the planet becomes self-sufficient.)
Lastly, we don't actually know what's far far out in space yet. We just have a pretty good idea, but there's a chance that we'll just be repeatedly killed by the unknown!
(60-120 years setbacks, unpredicted events, compensating for unknown)
Like I said, given enough time, anything is possible. But 250 years might be a bit too aggressive of a timeline. Given my estimates, I'd give us a minimum of 360 years; more realistically 500+ years. 360 is not too far from your window, though, so who knows? Maybe we can do it.