This is a common question and one you should have thought about before getting you chrome thingamajig.
I hope when you say Chrome OS you mean you have a Chromebook. In which case, irrespective of the fact that it runs a Linux variety, you won't be able to run any generic executable on it.
You could hack up a way for your executable to write result and get instructions from you cloud account but that is the best you will get, to the best of my knowledge.