So I was inspired by the legend Andreas M anotopotoolousposuts to flip a coin 256 times and generate my own private key. I will use this as my cold storage.
How do I then hash the 256 bits of data into a private
You don't. The 256 bits of data
ARE the private key.
and then public key?
The public key is not computed with a hash, it is accomplished with ECDSA using the Secp256k1 curve.
Is it possible to do the maths without a computer at all?
It would be rather tedious and complex to accomplish it with pencil and paper.
If not, is there some code I can run on an offline computer?
Convert to hexadecimal. Then you can run the code from
https://brainwallet.github.io/ on an offline computer and use your 256 bit value as the "secret exponent".
Alternatively, you can use paper and pencil to convert from binary to Wallet Import Format:
https://en.bitcoin.it/wiki/Wallet_import_formatThen import the resulting private key into an offline wallet such as Bitcoin Core or Armory.
The wallet software will take care of generating the public key and bitcoin address for you.