The problem is radiation. Empty space is not really empty, there are stray atoms floating around. Very scattered, but at a high impact momentum penetrate the ship ionizing anything in their way, and making the ship itself radioactive.

Not even talking about stray high-energy particles from distant supernovas and magnetars -- those irradiate ship regardless of its speed.

If "closest star" is Proxima Centauri, we're talking 4.5 light-years. Getting there in 1000 years means an average speed of 0.45% of the speed of light. At that speed less than 0.01 cm of titanium is enough shielding to keep the radiation out according to figure 1c) in my source below. Which makes sense because because this is ~10% of the speed of "normal" alpha radiation which is stopped by just your skin.

"Radiation Hazard of Relativistic Interstellar Flight" by Oleg Semyonov: https://arxiv.org/abs/physics/0610030 via Project Rho: https://projectrho.com/public_html/rocket/slowerlight3.php

anything going out there for those long journeys would/should be big, so there would be enough walls, storage of supplies, reaction mass tanks, etc. to provide significant protection.