I am currently using usb stick to transfer file to and from a cnc controller.
So, the CNC controller is a USB Host, and the USB stick is a USB slave. So is the Arduino.
I am wondering if I can use the Arduino as a usb mass storage
Which Arduino? The newer ones that do not use FTDI chips can be reprogrammed to appear as different types of USB devices to the PC. Presumably, they would then appear as the same kind of device to the CNC controller.
That is not a trivial change to make, though, and requires that you know how to make the Arduino respond to all requests from the USB chip for information. Learning the form of those requests and the expected form of the response is non-trivial.
Is this setup possible?
Possible? Yes. Likely? Easy? No.