From having been in the military for quite some time, yes, you can get a significant work out with only your body and the cold, muddy ground (and a good washing machine).
The bulk of a military work out is push-ups, sit-ups, and running, and very few of the exercises require anything but your own body and a firm surface. I actually recommend using soft carpet and a pad, especially for sit-ups, because hard ground can get painful after a while.
The real trick is to find enough variety to keep it interesting. There are at least a dozen different push-up variations that work out slightly different muscles, no-resistance upper body exercises (except the weight of your arms, which after a while, you realize can be quite substantial), and several varieties of sit-ups and crunches that can be switched out. Despite how it appears, there are also plenty of variation to running, such as sprints, running with weights (a heavy backpack, but make certain that it is evenly distributed), running backwards, etc.
The real fun comes, though, when you can find about a dozen people who all have active imaginations and don't mind yelling and having fun while they work out. Find some veterans who want to get back into shape, and you're guaranteed a good time.