Everything that I need to do to stay in shape at home can be done with a couch, dumbbells, towels, and a backpack, which nearly everyone has.
You can mimic the bench press for the chest, shoulders, triceps, and core with traditional push-ups. Then, you can mimic both incline and decline variations by either having your feet placed on the couch and pushing from the floor or your feet placed on the floor and pushing from the couch. Attack different angles — also increasing or decreasing difficulty depending on if you're mimicking incline or decline — by placing or pushing from the cushions where you sit, from the arms of the couch, or the back of the couch. For extra difficulty and as you increase in strength, grab the dumbbells, wrap them in towels, place that in the backpack, and put it on, tightening it up so it doesn't slide around.
Body weight squats to technical failure (where the next attempt would have you fall over) 5 times with a rest period of 1 minute 45 seconds to 2 minutes between sets will have your legs hurting the next day no matter what (especially if you are squatting with a dumbbell in hands or with the backpack setup). The Murph challenge includes 300 body squats for a reason. If you can get to 300 bodyweight squats in 5 sets or fewer, you're in good shape. I can't imagine him having done that with a "weighted" vest (flak with gear) on.
Then you can do deadlifts with the dumbbells. You might not have enough weight with dumbbells for this, so I would pre-fatigue first by doing some body-weight squats, then move on to deadlifting the dumbbells and you might feel it.
And then you have isolation exercises, which you can do with the dumbbells.
The only real thing that you'd need at home would be a treadmill or rowing machine for cardio. That's the one piece of equipment that you really can't mimic anything with, except for knee highs/running in place. And let me tell you, knee highs (raising your knees up, rotating left and right), sound easy, but after doing them for 45 minutes to 2 hours, with other things worked in like pushups, etc., they will get very difficult, very quickly, and could be your cardio.