Place potatoes in a large pot and cover with salted water; bring to a boil. Reduce heat to medium-low and simmer until tender, 20 to 25 minutes. Drain.
Cool potatoes until easily handled. Peel and slice into a large bowl. Cool completely, about 30 minutes. Add onion and radishes.
Whisk red wine vinegar, mustard, salt, and pepper in a bowl or cup. Drizzle in olive oil, while whisking, until well combined. Stir in chives. Drizzle over potato mixture and carefully mix in. Set aside for 15 minutes. Season with salt and pepper.