No, not space travel: Most of the bad effects are because of microgravity. We have no idea of the moon's or Mar's gravity is enough to alleviate those effects. But any space colony could rotate to create any simulated gravity we need.
The other source of possible health problems is the increased radiation in space. But with proper shielding, on a spacecraft, the moon, or Mars, that problem could be removed. The space station is NOT a very good shield because all the parts have to be light weight. A Mars or lunar colony would use underground spaces that would have great shielding. Any long-term space colony could use lunar or asteroid material as shielding.
