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!