Your fitness strategy should have 3 equally important parts:
For mobility you only need a yoga mat as equipment and maybe some elastic bands. For cardio you need a pair of good jogging shoes or a bike.
When it comes to strength you do need some heavy equipment. However this do not need to take up a lot of space.
The exercises in the Starting Strength program are the most "bang for buck" strength exercises.
You can use a heavy dumbell to do goblet squats instead of backsquats.
Also you can do push-ups (with a weight plate) on your back instead of benchpress.
The biggest problem will be the deadlift which also happens to be the single most important strength exercise. For this you will need a barbell and a lot of weights.
I would think you need this:
- 2 x adjustable dumbells with up to 40 kg on one dumbell
- barbell + minimum your bodyweight in weight plates
Maybe its a good idea to get 50 mm (olympic) dumbells so that you can use the same plates for both the dumbells and the barbell.
A shorter 10 kg barbell will probably suffice unless you are very strong.
Elastic bands are nice for mobility and physiotherapist style rehab/prehab but useless for strength alone.