When I visit a website which exploits a vulnerability in my browser and I save this website to my computer and open later the offline version can my browser be hacked again?
Yes, but not always.
It depends on how the exploit works. Many exploits contain a second stage payload which the first stage (contained on the website) downloads and then executes. This is done for multiple reasons. For example, the first stage may test to see which second stage is appropriate for that individual browser version, operating system, etc. It may also serve to make it possible to easily update the second stage payload without needing to modify the first stage. Of course, it's also entirely possible for the entire exploit to be contained on the malicious webpage, making it dangerous to open it later.