Gather canning equipment and heat water in a canner as directed at sunset.com/canningsteps
Heat vinegar, 3 1/2 cups water, the salt, and sugar together in a medium pot until boiling. Put a ginger slice, garlic clove, and chile in each of 6 pint-size canning jars. Pack jars with green beans.
Pour hot liquid over beans, leaving a 1/2-in. headspace at the top. Put lids on top and screw on rings firmly--but don't force.
Process jars 10 minutes. Check that jars are sealed (see sunset.com/canningsteps) and store in a cool, dark place up to 2 years, or chill any that didn't seal; they'll keep up to 1 month.