1. Place cranberries in a bowl. Cut fennel into quarters and remove tough core. Thickly slice each quarter and add to bowl. Add apples and lemon juice and toss to combine.
2. Pulse 1/3 of mixture with 1/2 cup sugar in a food processor, stopping and scraping down sides of bowl often. Transfer to a large bowl. Repeat twice with remaining mixture and sugar. Cover and refrigerate until very cold, at least 2 hours. Serve cold. (Can be made up to 5 days in advance. Keep tightly covered and refrigerated. Periodically drain off any excess liquid.)