From the Sputnik rocket's Wikipedia page, the specific impulse of the boosters and the engine are both ~307 s. Low Earth Orbit requires a Delta V of ~9400 m/s. The mass of Sputnik is ~85 kg. Plugging these into the rocket equation gives an initial mass of just about 2000 kg. But the actual Sputnik rocket had a mass of 267,000 kg. Obviously, the engine, tanks, and all the engineering takes up mass, and it's likely that the Soviet Union wasn't operating with full efficiency, but nonetheless it's a massive discrepancy. Where does it come from?
The payload isn't the dry mass of the rocket that appears in the rocket equation.
The dry mass is the payload plus the tankage, engines, skin, guidance system, payload deployment system and so on—basically everything that isn't thrown out the back of the rocket.