Paper wallet related question: could I not just write down any 2^256 bit number I can think of to create a private key? If I can, why is this never done? And would I necessarily need software to hash out the public key, or could this also be done offline by hand?

Bitcoin public/private key generation: Can someone explain why it is infeasible to grab a pencil and a sheet of paper and create a private key by jotting down my favourite numbers and letters? Wouldn't any 2^(256) number be a valid key? And if it is, what is it about the hash function that makes it impossible to generate a public key from my made-up private key by hand?