Peel the carrots and cut them into sticks 7 cm / 3 inches long and about 1 cm thick. Trim 5 mm off the ends of the cucumbers and cut them into spears the same size. Halve or quarter the radishes so the pieces are about the same size.
Divide the garlic, dill, mustard seeds and peppercorns among 4 clean 350 ml / 12 oz jars, using 1 clove, 1 sprig, 1/4 tsp mustard seeds and 1/4 tsp peppercorns per jar. Stand the carrot and cucumber sticks upright and tuck the radishes in the gaps. Pack tightly, stopping 1 cm / 1/2 inch below the rim.
Combine the water, vinegar, salt and sugar in a 1.5-quart / 1.4 L saucepan. Bring to a simmer over medium heat, stirring for 2 to 3 minutes until the salt and sugar dissolve. Remove from the heat.
Let the brine cool for 10 minutes. Don't pour it boiling hot, because it softens the cucumbers and you'll end up with limp spears. Pour it over the vegetables until they're covered, tap each jar on the counter to release air bubbles, and screw on the lids.
Refrigerate for at least 4 hours before eating. The brine turns pink as the radishes bleed, and the flavor deepens by 24 hours. Keep the jars cold and eat within 2 weeks.