Bring the vinegar, salt, sugar, garlic, jalapeño, caraway seeds, and 1 cup water to a boil in a medium saucepan. Simmer until the salt dissolves, about 1 minute. Remove from heat and let sit for 15 minutes.
Pour over the fennel, cucumbers, carrots, and radishes in a large bowl. Cool completely, stirring occasionally, about 2 hours. Refrigerate in a jar for up to 1 week.