Inspect six 1/2-pint jars for cracks and rings for rust, discarding any defective ones. Immerse in simmering water. Wash new, unused lids and rings in warm soapy water.
While the jars are bring sterilized, stir together vinegar, water, and salt in a large saucepan. Add garlic and bring to a rolling boil over high heat.
Trim green beans to 1/4-inch shorter than the jars.
Remove jars from simmering water. Place 1 sprig of dill and 1/8 teaspoon red pepper flakes in each jar. Pack in green beans standing on end.
Ladle boiling brine (leave out garlic) into the jars, filling within 1/4-inch of the tops. Seal jars with lids and rings.
Place a rack in the bottom of a large stockpot and fill halfway with water. Bring to a boil and lower jars 2 inches apart into the boiling water using a holder. Pour in more boiling water to cover jars by at least 1 inch. Bring to a simmer (do not boil), cover, and process for 10 minutes.
Remove jars from the stockpot and cool to room temperature, 8 to 12 hours. Test jars for a good seal by pressing on the center of the lid — it should not move.
Let green beans ferment for 2 to 3 weeks in a cool, dark space before eating.