Space is possibly the harshest environment there is for living things, especially for more complex vertibrates such as primates. It requires a lot of genetic information to make a petunia, and at least an order of magnitude more to make a human being.
The hard radiation in which the universe is awash cuts through DNA strands like a hot knife through warm butter, causing cancers, sterility, and the potential for offspring to bear serious mutations.
But problems with radiation and the harmful effects of long periods spent in micro-gravity can be overcome with creativity and engineering.
The radiation problem can be beaten with magnetic plasma shielding. (VERY cool technology) Also, building the ship with a double-hull and filling the gap with water would help to shield the crew.
As far as the loss of bone and muscle tissue, that can be easily dealt with. Just put a spin on the habitat module and that will generate artificial gravity in the form of centrifugal force.
The real problem lies in finding a method of propulsion that would minimize the time spent in interplanetary transit. Our present chemical rockets' specific impulse just doesn't have the long-lasting punch we need. Also, ion propulsion can run for extraordinarily long periods of time, but acceleration is too weak for any manned mission. It's going to have to be some form of nuclear power - fission, fusion, or some combination thereof.
Maybe for interplanetary expeditions, but for interstellar travel, rockets won't cut it.