Within our solar system and possibly even further it just doesn’t seem possible to envelop the sun at all.
You can sum masses of all inner planets except Earth and Moon, divide by average density, set sphere thickness to 1m and find that surface area for Dyson sphere made from inner planets is approximately 10x of Sun surface area. So yes, you can cover Sun in way that blocks all sunlight from the rest of Solar system, using only inner planets except Earth and Moon.
Moreover, you actually don’t need to cover all of Sun. You need to cover only fraction of it which reaches Earth, which is hundreds times smaller.
You can sum masses of all inner planets except Earth and Moon, divide by average density, set sphere thickness to 1m and find that surface area for Dyson sphere made from inner planets is approximately 10x of Sun surface area. So yes, you can cover Sun in way that blocks all sunlight from the rest of Solar system, using only inner planets except Earth and Moon.
Moreover, you actually don’t need to cover all of Sun. You need to cover only fraction of it which reaches Earth, which is hundreds times smaller.