can space travel make you sick?

yes.

Constant changes in acceleration plays havoc with your balance. "Weightlessness" is really an impression of falling (that is why they call it 'free-fall').

Muscles that don't get used atrophy.

Over long periods, since the bones do not carry any weight, they lose calcium.

maybe at first

Absolutely!

Our physiology is not suited for zero gravity environment.

If The Gravity On The Space Ship Is Switched Off You Will Start To Feel Weightless And Sick.

If The Gravity Is On You Will Be Fine.

Yes. The threat of exposure to cosmic radiation can cause cancer, or other radiation related illnesses.