I prefer public gyms by far but you can build a very capable home gym that fits under half a bed. Forget squats, deadlifts and bench presses and bring in unilateral variations at higher rep counts and you won't need much more than a handle and a pair of 45's for years.

I just measured the footprint of my home gym (only weights, a small bench and a pullup rack) and it's like 2 (two!) square meters. Take another 2 square meters for exercising and lifting, and I'm good to go.

Caveat, I parked my bench inside my pullup rack (don't have to move it before training though).