There is evidence that outer space radiation would have severely affected the astronauts
Apollo missions so far have been the only manned mission beyond Earth low orbit. In order to reach Moon space-ships had to cross Van Allen belts. The crossing could have been lethal, or at least would have permanently damaged the astronauts. Most astronauts did not develop health problems later on that could be related to that.
The level of radiation of the Van Allen belts can be crossed with proper shielding. Most probes and satellites do this on a regular basis.