If you take humanity's current energy and technological capacity and project a steady increase into the future, the chances of us reaching the stars any time soon look bleak. Even our nearest stellar neighbor is at least 300 years away.
Of course, humans could technically reach Alpha Centauri long before the 2300s. The Voyager probes, for instance, are headed into interstellar space, and it wouldn't be that difficult to take another probe and aim it in the rough direction of Alpha Centauri. But that probe would still take a long time to get there, its instruments would probably give out long before it arrived at the star, and what data it could provide would be hugely limited.
On the other hand, an Alpha Centauri probe that could reach the star within a human lifetime (say, 75 years), could actually assume an orbit around the star to maximize the data it could collect (which would be considerably more complicated than a simple flyby), and possessed sufficient communications ability to relay its data back to Earth would probably have to weigh at least 10,000 kilograms. That's about 13 times the mass of the Voyager probes and 20 times the mass of the New Horizons probe currently headed out to Pluto.
And that's the big problem. It would take a lot of energy to build and launch such a probe, and it's questionable how much scientists could actually consume in the building of the probe. According to new calculations by astronomer Marc Millis, the maximum percentage of its total energy the US devoted to spaceflight in any given year was only about a millionth of its total supply. That's not a lot of energy to play around with when launching such a sophisticated interstellar probe.
In fact, based on his estimates for the world's energy growth, the earliest possible launch for an Alpha Centauri probe is 2247, with the average launch date around 2463. And even that presumes some breakthrough in spacecraft propulsion. If we're stuck with what we're using now, then we couldn't launch the probe until 2301, with the more likely date some time around 2566.
Obviously, these are only rough estimates, which Millis himself readily admits. But they're worth considering in terms of how much energy it will take to get to another star system, even it's just a reasonably robust probe. For more on these numbers, check out our friend Paul Gilster's analysis over at Centauri Dreams.