Combine sugar and water in a saucepan over medium heat until just starting to simmer. Remove from heat. Cool simple syrup down to room temperature while preparing the other ingredients.
Slice celery into small pieces; transfer to a blender. Add salt, lime juice, and the cooled simple syrup. Blend until smooth. Pour mixture into a strainer set over the saucepan you used for the syrup. Press mixture through the strainer using a spoon until all the juice has been extracted. Cover and chill juice until thoroughly chilled, at least 1 hour.
Pour mixture into an ice cream maker and freeze according to manufacturer's instructions, about 20 minutes. Cover sorbet with plastic wrap, place a plate over the bowl, and freeze until set, about 4 hours.
