The very bulky suits you saw are those for an EVA, but the Spacex suit is to be used only inside a spaceship. The suits used outside the ISS or on the Moon need a micro-meteorite protection and a cooling system to remove excessive heat from the suit.
The suit shown in the link from JCRM, used for the Mercury project, was not bulky at all. It was the very first space suit worn by American astronauts. But it could be used only within the Mercury capsule and not in space for EVA.
The suits used for Gemini and Apollo were designed for EVA and look more bulky.
The SpaceX suit is designed for very short ballistic flights; there is no liquid cooling undergarment and no micro-meteorite protection. It is to be used only inside the spaceship and not in an EVA. The suit shown in the news was not pressurized. It would look less slim when worn in a vacuum with operational pressure inside.
No very special materials are needed to build a suit looking like a SpaceX suit. But you can not build a suit for EVA use in this design.