Given current technology, how far could we send someone before they get a lethal dose of hard radiation? What if they don't need to make a return trip? I know that last I had heard, Mars was too far to be able to take a return trip without being exposed to too much radiation, but it's possible that technology has improved since then.
In order to properly assess this, you need to take in to account a few things.
Much of my answer to this was taken with help from The Case for Mars, by Robert Zubrin.
First of all, let's start with the long term limits. As a starting point, let's look at the legal limits according to OSHA and NASA. These are defined in the following table, taken from NASA. These limits provide for an exposure per year of 50 REM, and a career of 100 to 400 REM, measured internally. To simplify this, I'm going to assume that the trip is a return trip, specked at 50 REM per year, and a 45 year old male is making the trip. That gives a 6 year mission, assuming we can keep the radiation exposure below 50 REM per year, and longer given reduced portions, such as while on the surface of a planet.
In the Case for Mars, they site an average cosmic ray dosage of between 20 and 50 REM per year, depending on where the sun is at during its cycle. The minimum take place when solar activity is at it's max, as the sun isn't blocking the rays (They come from outside the solar system). This is the primary source of day to day radiation exposure. This is below the limits easily.
For the second part, you need to consider rare high energy events, namely Solar Flares. If you were outside of a spacecraft with no shielding these events could be quite deadly in a short amount of time. However, we do know a few things about these. First of all, there is always advanced notice of them coming, at least in the range of hours. Secondly, the primary source of energy is charged particles, which can be repelled with a modest amount of shielding.
The typical plan for such storms is to have a specific shelter for these storms. Hydrogen has been found to be effective in absorbing the radiation quite well. Basically, if one hides out in the pantry, one would be relatively safe. One could do so for the few days required to wait out the solar storm, and the radiation would be reduced to a manageable level.
Bottom line, a 45 year old man could take a 6 year trip in space, or a 45 year old woman could take a 5 year trip in space, while meeting all OSHA requirements for astronaut safety. That could enable a person to make a trip to Jupiter, the Asteroid belt, and everything closer, without any major concern of radiation.