Short answer: Yes, you can get a virus just by visiting a site in Chrome or any other browser, with no user-interaction needed (video demonstration). Even with Chrome you are not 100% secure - and you probably never will be with any browser, but Chrome is getting pretty close to it and the security research community seems to agree that at this time, it is the most secure browser you can use.
Chrome, at this time, is the most secure browser that exists in windows, because of the sandboxing techniques it uses which add up to the security.
A good description of this sandbox is here: http://blog.chromium.org/2008/10/new-approach-to-browser-security-google.html although a bit dated. A more recent and technical one is here: http://dev.chromium.org/developers/design-documents/sandbox
The general idea is that a malicious website will have to use two separate exploits to achieve code execution on your pc: The first one exploiting the browser, the second exploiting the sanbox. This has been proven a very hard thing to do - it has not be done ever.
Your question comes at a nice timing: A week ago the security research firm VUPEN claimed that it broke chrome sandbox and published the following announcement: http://www.vupen.com/demos/VUPEN_Pwning_Chrome.php
If you watch the video, you will see that no user interaction was necessary besides visiting the malicious URL, and a sample application popped up out of nowhere (it could be virus instead).
The attack description proved to be wrong (but that does not matter much to simple users): The sandbox itself was not breached. It turned out that VUPEN exploited a bug in Adobe Flash Player, which is a plugin almost everybody has installed - and this plugin was not sandboxed. The answer of Google is that they are moving and developing fast so that this plugin will be sandboxed eventually in the next versions.
To sum it up: As you could see in the video, you can never be 100% sure that you are safe, so be careful, don't open links you don't know where they go or don't trust the sites.
Sidenote: NSA recently published a document indicating "Best Practices" for user security on the internet: Among those is the recommendation to use a browser that has a sandbox.
Here is the report: http://www.nsa.gov/ia/_files/factsheets/Best_Practices_Datasheets.pdf
Another sidenote: Internet Explorer 9 is advertised as having a sandbox, but security researchers agree that it is not properly implemented and have demonstrated successful attacks against it.
And a third one: The incognito window has nothing to do with virus protection, but can be useful in certain classes of attacks where cookie stealing or similar is the target because it separates browser instances and isolates information available.
Finally, there are technologies that offer better protection, like sandboxie and running browsers in virtual machines.