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At a distance of 4.367 light years from Earth, Alpha Centauri is the closest solar system outside of our own. With conventional rocketry (if it could even be done) it would take over 6000 years to reach. In fact even if a craft was built able to travel at 1/10th the speed of light it would still take 44 years to get there. Actually, if you wanted to stay and visit Alpha Centauri it would be more like 60 years to get there as half way there you would have to start slowing down so as not to blow buy it!
So is there another way. A way with current technology, resources and knowledge to build a ship capable of reaching, visiting and returning from Alpha Centauri in the span of a normal human life (75 years)? There are lots of theories out there, like Laser propelled solar sails for example. The problem with that type of idea is there is no way to stop once you get there, or return if you could stop. Thus it would be a one way mission and reserved solely for robotic craft. Furthermore, any communication from that robotic craft back to Earth would take decades. It seems to me there is actually more than one issue at hand when it comes to Interstellar Space Travel.
Problem 1: Propulsion Systems
The only feasible way to send humans to Alpha Centauri would mean either a) developing a means of propulsion that can reach speeds of at least 1/4 the speed of light (20+ years each way) or b) developing a means of traveling faster than the speed of light or c) developing a means of shortening the distance between Earth and Alpha Centauri (folded space, worm hole, etc.).
The most likely, within any reasonable time frame, would be developing a means of propulsion that can reach speeds of at least 1/4 the speed of light. However, even if that were possible it would take 5 years to reach that speed, about 13 years of travel at that speed and another 5 years of braking to stop in Alpha Centauri making the trip at least 23 years each direction. A 46 year round trip would require massive amounts of food, water, oxygen, supplies, etc. even if there were only 1 or 2 people making the journey. At least 1 gallon per day of water per person would mean they would need 34,000 gallons of water at a weight of 204,000 pounds. You add in food, oxygen and everything else they would need and just in people and supplies you are over 500,000 pounds. Then add in a few hundred thousand pounds for the ship itself, propulsion systems, braking systems, etc. and you could easily top 1 million pounds. That much weight would likely mean it would take 10 years to reach 1/4 the speed of light, 13 years travel at 1/4 the speed of light and 10 years to brake. Making the trip 33 years each way, which would mean more supplies would be needed. Thus the feasibility of sending humans on such a long trip is almost nil. Unless a form of suspended animation was employed and the ship was robotic during the most of the flight.
That all but rules out option a) unless, of course, we could travel at half the speed of light or faster and reach that speed in months instead of decades. Even then you would be looking at nearly 15 years each direction.
So how about faster than light travel? Given our current understanding it is impossible to accelerate up to the speed of light as doing so would require infinite energy. However, it might be possible to instantly jump from 0 acceleration to speeds faster than the speed of light. CERN recently managed to accelerate Neutrinos faster than the speed of light. Now Neutrinos have no measurable mass but it still opens the door of possibilities! Recent experiments in Quantum Physics have shown that particles (of matter) that are quantum entangled act oddly. In essence, anything that happens to one of the particles instantly happens to the other regardless of how far apart the particles are. Imagine if we could build a propulsion system that relied on this effect between quantum entangled particles to produce propulsion. If what happens to one particle instantly happens to another over a distance couldn't we use that the generate thrust instantly? Imagine a long tube (say half a mile long) that runs through the entire ship. At the front of the ship is a convex dome protecting the front of the ship. At front end of the tube is a bunch of particles of fissionable material (like Plutonium). A half mile away at the rear end of the tube is a concave dish with a particle cannon for lack of better word that can eject those particles of quantum entangled material out the rear of the ship a millisecond before the matching particle at the front of the ship is detonated with a laser. The particle detonated at the front would explode against the convex dome and direct the force through the tube towards the rear of the ship generating propulsion. Instantaneously a small particle at the rear of the ship would explode thanks the the quantum entanglement effect generating more propulsion. The reaction between the quantum entangled particles exploding half mile apart at the same exact time due to the quantum entanglement effect could quite possibly cause the initial acceleration to exceed the speed of light.
Even if we lack the ability to create massive amounts of quantum entangled fissionable material, know which particles are entangled to get exact detonations. The same design using normal fissionable material would reach speeds nearing the speed of light with a short period of time. If the Plutonium is in granules the size of a grain of sand and can be detonated precisely timed front and rear of the ship. Each single grain detonated should generate more propulsion that every rocket humans have ever fired into space. Imagine detonating 5 of these grains of sand at each end (10 total) over a 1 minute time span could propel a ship from Earth around the moon and back to Earth in a couple hours tops. If you could do a controlled detonation of 2 grains (one front and one rear) every 30 seconds for 1 day it would generate enough propulsion to make it to Mars and back to Earth in about a month!
Now detonating 2 grains every 30 seconds for one year then inverting the dishes at both ends to use them to slow the ship and detonating 2 grains every 1 day would get you to Alpha Centauri in roughly 10 years. Thus making it a 20 year round trip! In theory you could detonate 2 grains every 30 seconds for 3 years, invert the dishes and detonate 2 grains every 30 seconds for another 3 years and make it to Alpha Centauri in 6 years. Making the round trip in 12 years!
Problem 2: Communication
Even if we communicated via laser bursts back to an orbital station that was stationary (always facing Alpha Centauri) it would take 4.4 years for the transmission to reach Earth from Alpha Centauri. However, what if we could build a transmitter/receiver pair that utilized Quantum Entangled Particles. Changes to the particles in one transmitter/receiver would instantly happen in the other. If those devices were able to automatically recognize and decode the changes to the quantum entangled particles we would have instant communication between Earth and the ship regardless of where it was in the Milky Way Galaxy. Assuming we could build such a pair of devices it would allow for constant and instant transfer of information between the ship and Earth and make such a mission far more feasible. Especially if the devices had the capacity to relay large amounts of data like audio/video streams. The entire voyage could be live streamed to the internet and make trillions of dollars in ad revenue alone.
To summarize, I don't think the question should really be "Can we do it?" but more "Why aren't we doing it?"! We have the means, the technology and the resources. All we really need is some rich guy like Elon Musk to see it as an investment opportunity, put together a team to build the technology and ship and then pick the crew and send it on it's way with the entire voyage from Earth Orbit to Alpha Centauri and back live streamed to the internet for the world to watch!