We've done that. The ISS has been occupied for the last 25 years uninterrupted. The space station can go months between resupplies. In other words, we've shown that a crew can survive in a spacecraft without aid from earth for months. With some mission specific planning, I don't see why we couldn't manage a 6 month mission.
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Well the reason we don't have nuclear thermal rockets boils down to budget cuts at NASA and environmental/safety concerns around nuclear. We made significant progress on two different nuclear rocket designs before they were scrapped for entirely political/budgetary reasons. And by budgetary reasons I don't mean that the program proved to be too expensive or difficult, I mean that NASA's annual budget was year after year and they simply had to drop some projects.
Chemical propellants are great for launch, but the advantage of nuclear for deep space missions are really immense. The additional efficiency means you can make shorter trips, bring more supplies, and have more redundancy for equipment failures. It also provides the possibility of bringing the entire craft back home for future missions rather than simply expending it.
And as a power source, solar is fine around earth. But for trips further out, like to Jupiter, well at that distance your panels would only get about 4% of what we get here around earth... That's just not going to cut it for crewed missions.
Honestly, spacecraft are probably the absolute best use case for fusion power. They're one of the few contexts where the energy density is extremely important and the high cost is still worthwhile.