The right home gym set for you will depend on what you like to do and what you wish to achieve. For someone who loves yoga, a mat and large wall mirror might be enough.
For someone who likes to run, a treadmill is a must. Find a slim personal treadmill that can be folded away if you don’t have much space. Adjustable incline is a nice feature to have as well.
As any experienced runner will tell you, injury awaits if you don’t cross-train. You need to do other things to strengthen your body and improve your cardiovascular fitness without pounding your joints. Get a spin bike and some dumbbells.
If you are on a budget, then consider budget options just as resistance bands, jump ropes, suspension straps, Olympic rings, battle ropes, balance boards, stability balls, aerobic step platform, and the like.