You actually can, although I don't know how rugged the result is. You probably could make a heavy, one speed bike out of wood with like, wheels that are just big disks. I'm not sure if it would beat walking, especially before purpose built roads were common. That being said, they might at least think going down a hill at speed is fun, which is what the first bikes were made for.

For a modern-style bike, the wheels are more of an engineering challenge, as is centering the various parts and ensuring a tight fit. Modern machine parts are made with micrometric precision, which involves surprisingly simple tools, but a whole lot of science and technique.

If it was a few thousand years later after horses were introduced, they could copy the concept of tension wheels for their chariots.