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Can this technology Chrome App or Google NaCl provide me with this capabilities?
Basically it should be able to run terminal commands (even by asking permission first).
And it needs to be able to edit client's filesystem.
I am interested in Chrome App or Google NaCl because they both can run in any operating system (that supports chrome).
Chrome apps cannot execute arbitrary system processes (aka run terminal commands). There is no permission for this.
Chrome apps can communicate with others apps using message passing, sockets, etc, but you would need a special terminal application to receive these messages -- which would mean first solving the problem you are currently asking to solve!
According to Google, Chrome is supported on some Linux, Mac OS X and Windows.
As your need is not strictly related to Chrome neither a browser, I suggest you to use (or at least have a look at) Java, which is supported on much more environments any you can safely execute terminal commands. With or without authorization.