Andy, arcsecs are indeed a useful size for some objects but so are degrees. For some objects, the milliarcsec and microarcsec are the appropriate measures, especially in astrometry and VLBI radio.
This leads me to the proposal that degrees, millidegrees, microdegrees, nanodegrees, … would be the appropriate way of proceeding. Alternatively, and this I find rather more attractive myself, radians and their SI sub-divisions must be the natural units. You may claim that a radian is a rather large angle but the size of the farad has never seriously worried the electronics engineers who are quite happy working with pF capacitors.