Perhaps an expedition like the Robinson family towards a planet in Alpha Centauri system will happen some day, something in about one hundred years or more, but of course not in 1997! This storyline is perfect, because today we know that Alpha Centauri system of three stars is among the most likely candidates beyond the solar system to shelter life like ourselves. The two main stars (Alpha Centauri A and B) are very like our Sun in all of their features and they are the closest (about four light-years) from us. Unfortunately, we don't know still whether the stellar system has planets orbiting around it or not. Likely, giant planets like Jupiter or Saturn don't exist, because otherwise they would be discovered by our nowadays technology. However, the existence of smaller planets like the Earth could exist, because we don't have still the technology needed to discover them. By another side, the lack of giant planets would limit the time of existence of life, because the protection against the impact of meteors (like that one which has extinguished the dinosaurs in Earth) would be missing (we exist today due to the protection of giant planets like Jupiter). This is an answer that only time will say!
The only inconsistence in this episode, in my opinion, is to send only one family (a couple with three sons) and a man to colonize an unknown planet. The correct would be to send three families or more genetically different among them, in order to avoid troubles with later offspring. The spaceship must be much greater than the Jupiter 2 for this purpose as well!