That’s what the people at the MIT Game Lab are trying to answer with A Slower Speed of Light. You can go download and play it for free at that link, and I highly recommend it.
Basically, this is mostly a demo to show us what happens to vision and movement as we approach the speed of light. The way the game works is that there are 100 easily accessed orbs to collect on the board, but every time you pick one up, the speed of light is lowered, bringing you closer and closer to that speed. As you approach the speed of light, it changes how visuals are processed on the screen and how you move relative to the objects around you.
Mostly, this is a stab at creating a game engine that others can use, but doing What’s It Like to Play a Video Game Close to the Speed of Light? it as accurately as possible to the actual experience of moving in a familiar place, but closer to the speed of light.
Moreover, it’s a fun little thing to play with for a few minutes. So go play with it already.
(h/t Starts with a Bang)