You guys got me to thinking on this so I went and did a little further study.
A question was asked on the sugar being white, apparently from what I read anyway the main way of removing the color is by putting chalk into the juice mixture. The chalk binds with the non sugar chemicals and to itself creating balls of chalk within the mixture and thereby removes much of the color and other non sugar chemicals. From talking to people who worked at the sugar plant in Nyssa there are also some chemicals they use to whiten as well.
On the number crunching end of it:
A Co op in Minnesota, 50,000 acres of beets planted produced 768,000 tons of beets, that divides out 15 tons per acre or ".7 pounds of beets per square foot planted."
The sugar plant they made in this Co op had a record day, sliced 8,873 tons of beets (17,746,000 pounds) to produce 2,174,600 of sugar.
On a commercial scale going by this days production it takes around "8.15 pounds of beets to produce 1 pound of sugar."
To produce one pound of sugar one would expect to plant around 12 square feet of sugar beets.
To produce a 25 pound bag one would plant around 17.5 ft by 17.5 ft for about 300 square feet.
1,000 square feet, 32 ft by 32 ft would produce around 85 pounds of sugar.
These numbers are based on commercial efficiency, on the growing end with a small crop you may well be able to outdo the commercial crops, the production of sugar on a small scale you may or may not be able to get the efficiency of the commercial system.
If you have an interest in producing your own beet sugar you now have some numbers to work with, like all numbers they are very relative, but should get you in the ball park.