Cut grapefruit in half; cut each half into 1/2-inch slices. Place grapefruit in 1-quart or larger glass jar with tight-fitting lid. Pour vodka over grapefruit, completely covering with vodka. Cover with lid. Let stand 4 days in cool, dark location.
Add tarragon sprigs; cover and shake well. Push tarragon into jar using spoon or fork so it is completely covered with vodka. Let stand in cool, dark location at least 2 more hours but no longer than 2 days.
Line strainer with cheesecloth; pour vodka through strainer into clean bottle. Cover; store in refrigerator up to 6 months. Discard grapefruit and tarragon.