Can you build real strength at home without any equipment?

Yes, meaningfully, though not indefinitely. Bodyweight training builds genuine strength for a long time if you progress the difficulty rather than just adding reps — moving from knee push-ups to full to elevated to one-arm progressions, or from assisted squats to split squats to single-leg. Fifty easy reps is endurance work; six hard ones build strength. The honest limit is lower body, where bodyweight eventually becomes too light and you need either single-leg variations or something to hold. For most people that limit is a year or more away.