Yes! Literally get down on the ground right now and do a set of pushups. Congrats, you just did your first workout, it took less than a minute.

Do one set of pushups each day for the next week (one set of pushups for me is 30-40 pushups in a row, I basically keep going until I can't anymore). After the first week, add in another exercise, situps are a good one. And the week after that, get a pullup bar or some dumbbells and add in another exercise. Keep adding a different exercise until you have 5 or 6. Do one set of each exercise with little to no rest in between each exercise. This should take less than five minutes each day. At any point if you fall out of habit, go back to square one, i.e. one set of pushups each day.

Once you have made this into a regular habit, you can start adding more reps or add a second workout (i.e. one in the morning, one in the afternoon). Continue to add more reps/exercises/workouts until you feel satisfied with the routine. For me, I do one set of 6 strength exercises per day. It takes me less than five minutes each day. Keep in mind this is just my strength routine. I also run, hike/walk, bike or surf 3-4 times a week. If I wasn't already in the habit of doing that, I would add some cardio to my daily workout (jumping jacks, up/downs, etc...)

I wouldn't even consider getting a gym membership unless you have a very specific need for a certain routine or set of equipment. For example some folks like to mix in cardio on a machine (rower, elliptical, etc...) with weight machines in a single workout. Keeping all that gear at home would be tough. That is fine, but just remember that getting to and from the gym requires a time and effort, and that can result in losing the habit. So only go this route if you either have a very specific need that you can't replicate at home, or if you know you won't drop the habit.