The belief that you need a gym to get fit is simply not true. Calisthenics athletes demonstrate that bodyweight training, when properly programmed with progressive overload, can build extraordinary strength and muscle. The key is understanding how to make exercises progressively harder as you get stronger - the same principle that drives progress in the gym.
A complete home workout programme needs to address all movement patterns: push (push-ups and variations), pull (rows and pull-ups if you have a bar), hinge (Romanian deadlifts, good mornings), squat (bodyweight squats, jump squats, pistol squat progressions), and core (planks, hollow body holds, leg raises). Cover all five movement patterns consistently and you'll develop balanced, functional fitness.
Progressions for home exercise: push-ups progress from wall push-ups to knee push-ups to full push-ups to feet-elevated push-ups to archer push-ups to one-arm push-ups. Squats progress from box squats to bodyweight squats to jump squats to Bulgarian split squats to pistol squats. Pull-ups progress from table rows to band-assisted pull-ups to full pull-ups to weighted pull-ups. At each stage, add reps before advancing to the harder variation. Train 3-4 days per week with this structure and you'll achieve visible results within 8-12 weeks.