Using nuclear propulsion, as proposed fifty years ago, would take less than 50 years. Only relatively modest engineering problems need to be solved, without the development of exotic drive systems or warp speed.
https://en.m.wikipedia.org/wiki/Project_Orion_(nuclear_propulsion)
Nuclear Thermal and Nuclear Electric Propulsion (NTP/NEP): Another possibility for interstellar space flight is to use spacecraft equipped with nuclear engines, a concept which NASA has been exploring for decades. In a Nuclear Thermal Propulsion (NTP) rocket, uranium or deuterium reactions are used to heat liquid hydrogen inside a reactor, turning it into ionized hydrogen gas (plasma), which is then channeled through a rocket nozzle to generate thrust...
However, despite these advantages in fuel-efficiency and specific impulse, the most sophisticated NTP concept has a maximum specific impulse of 5000 seconds (50 kN·s/kg).
...adjusted for a one-way journey to Proxima Centauri, a nuclear rocket would still take centuries to accelerate to the point where it was flying a fraction of the speed of light. It would then require several decades of travel time, followed by many more centuries of deceleration before reaching it destination. All told, were still talking about 1000 years before it reaches its destination. Good for interplanetary missions, not so good for interstellar ones.
Nuclear Pulse Propulsion: Nuclear pulse propulsion is a theoretically possible form of fast space travel...
But of course, there the inevitable downsides to the design. For one, a ship of this size would be incredibly expensive to build. According to estimates produced by Dyson in 1968, an Orion spacecraft that used hydrogen bombs to generate propulsion would weight 400,000 to 4,000,000 metric tons. And at least three quarters of that weight consists of nuclear bombs, where each warhead weights approximately 1 metric ton.
All told, Dyson’s most conservative estimates placed the total cost of building an Orion craft at 367 billion dollars. Adjusted for inflation, that works out to roughly $2.5 trillion dollars – which accounts for over two thirds of the US government’s current annual revenue. Hence, even at its lightest, the craft would be extremely expensive to manufacture.
There’s also the slight problem of all the radiation it generates, not to mention nuclear waste. In fact, it is for this reason that the Project is believed to have been terminated, owing to the passage of the Partial Test Ban Treaty of 1963 which sought to limit nuclear testing and stop the excessive release of nuclear fallout into the planet’s atmosphere.
Fusion Rockets: Another possibility within the realm of harnessed nuclear power involves rockets that rely on thermonuclear reactions to generate thrust...
But of course, the project also identified numerous stumbling blocks that made it unfeasible using then-current technology – most of which are still unresolved.
For instance, there is the fact that helium-3 is scare on Earth, which means it would have to be mined elsewhere (most likely on the Moon). Second, the reaction that drives the spacecraft requires that the energy released vastly exceed the energy used to trigger the reaction. And while experiments here on Earth have surpassed the “break-even goal”, we are still a long way away from the kinds of energy needed to power an interstellar spaceship.
Third, there is the cost factor of constructing such a ship. Even by the modest standard of Project Daedalus’ unmanned craft, a fully-fueled craft would weight as much as 60,000 Mt. To put that in perspective, the gross weight of NASA’s SLS is just over 30 Mt, and a single launch comes with a price tag of $5 billion (based on estimates made in 2013).
More: http://www.universetoday.com/15403/how-long-would-it-take-to-travel-to-the-nearest-star/