So if you're not a smarty pants know-it-all like Bob, another way you can estimate the size is with the lens correction filter in Photoshop. Turn the grid on in the preview window to make it easier, then adjust the settings until your sign looks like a rectangle. Then scale the photo in Illustrator based on something in the image that you can estimate the size of. Obviously this is only useful for estimates - you almost always have to measure on-site before production.
Yes, I know you said you don't have Photoshop, but that's how we do it here. I thought it would be worth explaining for anyone else reading this thread that does have Photoshop, or maybe whatever software you use has a similar tool?