Another place to consider, which would probably be more feasible, is the Atacama Desert in northern Chile, near the border of Peru/Bolivia. More specifially, go to the town of San Pedro de Atacama -- there's tons to do from there, as it's kind of the "base of operations" for the area. It's quite remote, but popular with backpackers.
I went there in July 1995 and stayed at a tent camp, which was really cheap. We took a tour that took us to El Tatio geysers, then to some thermal baths, and a stop by the salt fields (it's called something like that -- don't quite remember the name). You can also go horseback riding, and the atmosphere of the place (at least when I was there) was friendly and fun.
The best part of our stay in the Atacama involved taking a tour out to Valle de la Luna, a meteorite crater. Instead of going back with the group, though, we pitched a tent and stayed overnight. A truly stunning, breathtaking experience...
Atacama isn't far from Macchu Picchu, relative to Chilean Patagonia (which is on the far other end of the country), plus you don't need many days to enjoy it. However, it's a unique place, so you won't feel as if you missed out on something. I'm sure all you'd need to do is get a bus going to Arica, and from there transport to Macchu Picchu.
