No one needs any equipment. Good old pushups, crunches, wall squats, climbing stairs, running/walking, and planks will all do great things for you. Yoga will help too, and you don't NEED a mat or expensive clothes to do yoga. If you want to add weight for resistance, find things around the house - a gallon of milk, cans of veggies, etc.
I have equipment and a treadmill, but 70% of my routine is "old school" stuff. Good luck!
Couldn't have said it better myself.
Another trick is if you have furniture movers/sliders - use those to make lunges, rollouts, mountain climbers, planks, etc... more challenging.
Also, you just need to piece together maybe a couple groups of 2 or 3 exercises to make your own circuit. Then change up the combinations every couple weeks so your body doesn't get "bored" or used to a routine - making the exercise less effective.