You'd be better off with a 690 perhaps...
It can be done on the 500, but it really isn't designed for long miles and hauling lots of luggage. So you'd be spending a good bit of money updating the lighting to make it minimally safe for night riding (because no matter what people say, you WILL get stuck riding at night on an adventure ride
). It will need a bigger tank, at least 5 gallons. My 530 carries 6.6 gallons. The seat is not good for many miles, unless you are standing for most of them. Luggage capacity is limited because there really isn't much of a subframe for attaching and supporting bags. I have the Kreiga Overlander 30 bags on my 530 and a top bag as well. Total weight for the luggage is maybe 60-70lbs and I would NOT want to go much over that at all. That is a minimalist load as well, carrying very little food. I have a front/rear tube, small chair, sleeping pad, tent, sleeping bag, ONE shirt and those real thin pants you can wad up and zip the legs off to make them shorts, some tools and spare oil. That is about it. If you are used to camping off a big bike like the GS, you will have to do some SERIOUS rethinking of what you need to take and how to pack it. Then, even if you get it down to something similar to what I carry, the weight will still affect the handling of the bike if you try to ride in anything really technical. The rear bottoms out a lot unless you upgrade the rear shock for the heavier load.
After two trips (week long each) on the 530 with RSquared (him on his 450 with almost an identical load as mine), we've come to the conclusion that it is better to have a base camp where you can leave your stuff and ride with a minimal load. We did that for half a day on our recent trip to Arizona and it made a BIG difference in the rough stuff!! Ideally, you'd do a relatively easy ride to a nice spot to camp that is central to doing several days of riding in the area. Not only do you get the weight off the bike for the fun stuff, but you don't have to setup and tear down camp as often.
The other issue is tires. When you get to the rough stuff you are going to want good tires. Those tires will not be so great for highway droning to get you somewhere, especially if you have to get across West Texas. So we've been loading the bikes and trailering to our destination, riding and camping for a week, then loading up and heading home. After a long hard week of riding, it is nice to ride in the truck for the haul home. It also leaves you fresher when you arrive where you want to ride and your tires aren't worn out from just getting there. Also, if we get 4-5 guys on the trip, splitting gas in a single truck for hauling out cuts down on fuel costs for the trip even with the lower mpg of the truck.
Dave's trailer comment with the GS is not out of the realm of possibility, but I would not want to leave my GS somewhere unattended while I was out riding the other bike, unless I had a real good place to leave it.
For a bike where you just head out, I would go with something like a DR-650, KLR-650, 800GS, Tiger 800, maybe a Vstrom 650/1000, or other similar and more street oriented bikes. If you run knobbies, bikes like the DR650 and KLR650 can get you through some pretty crazy stuff, but it is a LOT of work compared to a more dirt oriented bike unless you are just a real good rider. Gotdurt lead our recent Az trip and he was on a tired old DR650 with 20K+ miles on it. He out rode me and RSquared on our KTMs hands down! I think he is part mountain goat!