Is it possible to specify a hotkey that will activate a Google Chrome browser action?
No, you can manipulate almost every other aspect of the browserAction and the popup (including closing it) but it cannot be triggered programatically.
@hamczu is right that the only way to bind global keyboard shortcuts is to inject a Content Script that listens for keystrokes in every page.
However you will not be able to make those keystrokes (or anything else) trigger the browserAction.