The general problem is cumulative exposure to cosmic rays on long-duration flights. It's not clear if even a Mars mission has acceptable risk, and flights past

The general problem is cumulative exposure to cosmic rays on long-duration flights. It's not clear if even a Mars mission has acceptable risk, and flights past Mars mean everyone gets a very high risk of cancer and radiation-induced cardiovascular disease. To shield against the high energy component of cosmic rays, you need to wrap the crew compartment in something like a few meters of polyethylene or water. This is far heavier than anything we could launch today, but would work on something like a nuclear pulse rocket (that has a minimum weight requirement!) There's an additional problem around Jupiter, which is what Pioneer 11 discovered. The strong magnetic field causes lethal radiation levels at the inner moons. Again you can solve this by wrapping yourself in crazy amounts of water or plastic, but it's far beyond our capability now.