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)

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.

Let me put in a cautionary note about a potential problem that could arise. When I applied for my position in Germany, I was required to submit a Gesundheitszeugnis (health certificate) that was signed by an authorized doctor registered with the German embassy. I do not know how a diagnosis of cancer will play out in such circumstances.

However, if you are an EU citizen, this may very well not apply to you, because you will be able to relocate from one EU country to another through "freedom of movement."