I own an argon ion laser head with a beam diameter of '0.70' (using $1/e^2$ width).
What is the unit on the 0.70? And can it be converted to something more intuitive, such as meters?
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You'd need to refer to the technical specifications. For example, in the spec sheet found here, the beam diameter is listed as 0.70, and the note above the data table indicates that the measurement is in inches.
The $1/e^2$ refers to the way that the size of the beam diameter is defined. More information about this and the most common alternative measure of FWHM (full width at half maximum intensity) can be found here.