Is it possible to apply for a post doc (and get a post doc) while struggling with the health issues (a life-threatening illness: cancer) I struggle with the exact question (or at least will have to face it near in the future) that was asked here: Do I need to mention having survived a life-threatening illness when applying for a new academic position?

When I was a kid, I had a cancer. Now I work at the university as a research assistant and I'm on my way to a PhD. After finishing my PhD, I would love to apply for a post doc, preferably in Sweden, Iceland or Norway (but my advisor suggests US or Australia, so I'm not sure where exactly I'm going to go).

There is a serious possibility that I will have a reoccurrence  in few years (talked with my oncologist many many times about it), or will not have, nobody could tell that, but there is a strong possibility.

At my current university (middle Europe) it was not a problem, I wasn't obliged to mention it to anybody, however, I informed my employer about my illness. My previous condition is almost not relevant today, however, I do have days when I do not feel good and just want to rest and sleep (but have to go to the University anyway, and of course, I go).

So, with my condition, is it possible to find a university and apply to post doc or not? And another, not less important: do I need to mention about my health when applying for a post doc position?
Serious health problems can strike anybody at any time.  You know that you have a much higher than background risk of having your condition return, but you also have a pretty good chance of getting into a serious traffic accident or having a parent come down with a serious illness or getting hit by major depression or any of the myriad other ways that your ability to do your work can become impaired.  No sane organization expects its employees to be in a protective bubble where bad things cannot happen to them.

If you can do your work effectively now, and don't have a known crisis expected in the near future, then the only sensible thing to do is to carry on with life.  You don't need to inform potential employers, any more than you would need to tell them that you might get into a traffic accident because you commute.  As noted in the comments, though, you definitely do need to make sure you have good access to health care that will cover your condition if it recurs.