[mrep]

The 2 biggest health problems that I am aware of with spaceflight are due to weightlessness and radiation. However, both can be probably be solved with enough money (make the spaceship a rotating one with artificial gravity and add more shielding). There might be others but I am pretty confident you could engineer your way past each and every one given a large enough budget.

[jcranmer]

You're sort of missing the biggest health problem of all: how do you make a self-sustaining closed ecosystem?

[mrep]

True, I was thinking more in the context of the mars trip you mentioned where you can generally take the resources you need for the trip. For intersteller travel, you would need to make it self sustaining since the travel times are so long and that would be extremely expensive.

[maxerickson]

The article suggests moving stars.