[GeeTwo]

The two questions that feed into this are:
What is the physical size of the banner going to be? Certainly no larger than 10 feet square, because that would take up a full side of your pit. You should probably trim this a bit, because I understand some venues don't have the space to give you a full 10'x10' pit. If it will be in front of the pit, over the "door", you should limit it to about 3' tall so that it can be under the 10' height limit and allow tall people (and your perhaps 78" tall robot on a 6" high cart) to enter and leave easily. Figure your answer in inches.
The other question is what resolution (dots per inch) do you need? If you think people are going to walk up to the banner and read it at normal book distance, you'll need about 2000 dpi for photo quality, or 600dpi for a good (not photo) quality. If you're posting it on the back wall and no one will be looking at it from less than ten feet away, you can probably get away with 200dpi or even 100dpi. To answer this question, try creating a small patch (8"x10" or less) of what is likely to be the most critical piece of your banner, and print it at various resolutions, then look at it from your "nearest" distance.
Then, multiply your dpi number by the number of inches in your banner. For example, if you're doing a 3' x 8' banner (that's 36" x 96") that'll be on the back wall of your pit so you go with 200dpi, your image will need to be 7200 pixels by 19200 pixels - 138 megapixels!
However, if your design consists of large areas of constant color, you can do much better by creating a vector graphics file of some sort and sending it to the limner (sign maker) to execute. Find out what formats they can use before you start! Almost anyone can use a PDF or postscript file, and who knows, some of them may even be able to take your CAD output, or something you can export from CAD.