I'm not sure if this is exactly what you're asking, but there's a good science book called Cosmos by Carl Sagan that talks about a drive consisting of "small nuclear explosions against an inerial plate" providing a sort of "cosmic putt-putt."
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As noted, this was Project Orion, a serious but kinda sucky idea. Imagine having to explode dozens of nuclear bombs to get into space - not nice for those left on Earth. It's less bad as a propulsion method if you're already in outer space. It makes an appearance (in a launch from Earth!) in the Niven/Pournelle's novel "Footfall." I think of Project Orion as more of a
BANG-BANG than a putt-putt.
Adding to confusion is that one of the future NASA projects to go back to the Moon or to go to Mars (I forget exactly) is also named Orion. I about had a heart attack when I saw that name, I thought they were reviving the old nuke-bomb-propulsion idea and seriously considering doing it.
Assuming propellant is not a problem (lets say you can scoop up hydrogen and burn it), then there is no limit. At one G constant acceleration, you will, within months, be zipping from one star to the next in minutes. (This is ship time, of course.)
Giving the names for these things may be useful for those wanting to investigate further. This (an engine that sucks up interstellar hydrogen to use as fuel and/or propulsion mass) is called a Bussard Ramjet.
Well, how does a nuclear power plant work? You split an atom, and the resulting atoms around it all split and there's a big ass explosion that gets shunted somewhere and you've got electricity, right?
The atoms are split at a controlled rate so it doesn't heat up to a million degrees and blow up, but the splitting atoms (and resultant neutrons hitting things) creates lots of heat, which is used to turn water into steam to run a turbine which turns the electric generatir, just as in a traditional coal-fired power plant.
Couldn't you do the same thing but that energy is shunted out a big hole in the back of your ship? The explosion's force, then, exerts forward and backward, propelling the ship forward, as that's the equal and opposite reaction.
Right?
Right, and that's Project Orion.
There's also nuclear power propulsion that looks more like (part of) an electric power plant. Nuclear reactions create heat in the "bell" of a rocket motor, water is injected which the heat turns into steam which exits the rocket motor at high speed, causing the rocket to move forward. Since the nuclear material remains sealed up, this is much cleaner than Project Orion.
Here's a description:
http://www.astrodigital.org/space/nuclear.html
Regardless of how it's done, a rocket needs propellant to work, and that's going to be part of the payload (unless you do the Bussard Ramjet thing), so you always have to take that into consideration - it will limit how fast you can go (you can always go farther with the same fuel by coasting much of the way, and not going as fast).
I just want something relatively low-tech (near or at current tech) that's reachable within 20 years, if we'd get off our asses and do it.
I suppose either one of these (Project Orion or more 'traditional' nuclear rocket) is possible in that timeframe. The Bussard Ramjet is much more ambitious, and much less likely to be operational in 20 years.