Bears are a potentially serious hazard when running the Classic. There's never been any major incident but there have been close calls. Bears have eaten rafts, bluff charged racers and added a lot of excitement in an already dangerous race.
Pepper spray is not a bad option although in general pepper spray is more of a psychological protection than an actual defense against bears. A couple people have carried handguns. Adding several pounds onto your weight total by carrying a large caliber handgun is probably a bit excessive. Especially when you consider the abuse it will take during the race and how difficult it is to actually shoot a bear with a handgun.
Most racers rely on their wilderness skills to avoid bear encounters. This is probably the best thing to do. If you're running the race, hopefully you've already spent a fair amount of time in bear country. When you're racing you should be thinking like a bear anyway by taking the easiest route through brush and around hills on game trails. Avoid tall grass, salmon streams, and make noise when traveling through bear country. Sometimes you can smell bears. Some people say they can sense a bear is near by being alert and in tune with their surroundings. Whatever you do, just use your common sense.
Most racers will see grizzlies on most races. But most of the time, the bears will be going the opposite direction and you won't have to worry about it.