Put the turnips, beet and garlic into a wide-mouth, heatproof 1-quart jar.
In a small saucepan, bring the vinegar, salt, sugar and 1 1/2 cups of water to a boil. When the salt and sugar are fully dissolved, pour the brine over the vegetables to fill the jar. Leave to cool. When completely cool, cover the jar and chill for 1 week.