Stir together sugar and 1/4 cup water in medium heavy saucepan until sugar is completely moistened. Bring to boil over medium-high heat, 3 to 5 minutes, and cook, without stirring, until mixture begins to turn golden, another 1 to 2 minutes. Gently swirling pan, continue to cook until sugar is color of peanut butter, 1 to 2 minutes. Remove from heat and swirl pan until sugar is reddish-amber and fragrant, 15 to 20 seconds. Carefully swirl in 2 tbsp warm tap water until incorporated; mixture will bubble and steam. Pour caramel into 1 1/2 by 4 1/2-inch loaf pan; do not scrape out saucepan. Set loaf pan aside.
Adjust oven rack to middle position and heat oven to 300 degrees. Line bottom of 13 by 9-inch baking pan with dish towel, folding towel to fit smoothly, and set aside. Bring 2 quarts water to boil.
Whisk eggs and yolks in large bowl until combined. Add sweetened condensed milk, evaporated milk, whole milk, vanilla, and salt, and whisk until incorporated. Strain mixture through fine-mesh strainer into prepared loaf pan.
Cover loaf pan tightly with aluminum foil and place in prepared baking pan. Place baking pan in oven and carefully pour all of boiling water into pan. Bake until center of custard jiggles slightly when shaken and custard registers 180 degrees, 1 1/4 to 1 1/2 hours. Remove foil and leave custard in water bath until loaf pan has cooled completely. Remove loaf pan from water bath, wrap tightly with plastic wrap, and chill overnight or up to 4 days.
To unmold, slide paring knife around edges of pan. Invert serving platter on top of pan and turn pan and platter over. When flan is released, remove loaf pan. Using rubber spatula, scrape residual caramel onto flan. Slice and serve. (Leftover flan may be covered loosely with plastic wrap and refrigerated for up to 4 days.)