Is there any way that Chrome can be configured in order not to store browsing data for specific web pages?
No, not automatically.
You just need to remember to start an "incognito" window before browsing pages you don't want logging - as well as not sending cookies from non-incognito windows to sites this stops all permanent history and cache records too.
If you visit a site you don't want logged without going to an incognito window you can selectively remove the history records from the history screen. This does not remove locally cached files though, you can only remove them by clearing the whole cache or visiting enough other content that they expire from the cache automatically.