The model they came up with was a 570-foot behemoth called the Daedalus. Powered by nuclear fusion from 50,000 metric tons of helium-3 and deuterium fuel, it could get up to the speed of 21,277,777.8 meters per second, fast enough get from LA to Paris in a brisk 0.426 seconds.
Of course, at that speed if you hit a tiny dust particle, the damage would be catastrophic. To get around this, they gave the design a big beryllium shield to get rid of all the small specs, and (seriously) a big cloud of dust enveloping the craft that would obliterate virtually anything that got in its way. That's right, a goddam particle shield.
So Why Don't We Have One?
Even though all the technology called for in the design has been around since the 90s, the cost would be (forgive the expression) astronomical. Do you have any idea how much it costs to get your hands on 50,000 tons of deuterium/helium-3 pellets? A fucking lot.
Also, because of its enormous size, it would need to be constructed in orbit one bit at a time, and when it costs 10 million bucks to put one metric ton of crap into space, it would cost in the vicinity of $540 billion just to ship the raw materials up to the construction site. If only there were some way to get these things into space quickly and cheaply... like a cannon of some sort....
Hey, remember me? From before?
The other problem was that, in space, stopping is even harder than moving. This thing didn't really have brakes built into the design, so the plan was to instead have it take a bunch of pictures real fast while it zipped by the destination, beaming the images back to Earth as it went.
It then would zip through the system at Mach 64,000, presumably going on to crash into a planet, star, or unsuspecting alien.
In other words, we had something that would cost hundreds of trillions of dollars, require cooperation on a global scale, take forever to get there and at the end, we'd just have a few photographs to remember it. Basically it would be like an interplanetary version of a family trip to Disney World.
To further diminish your faith in science, check out 10 Famous Sci-Fi Weapons That They're Actually Building. Or take a look at The Insane True Story Behind the Birth of the Internet.