One of the Round 3 Finalists in the NIST post-quantum cryptography call is the Classic McEliece cryptosystem. Although it is one of the most secure cryptosystems, the large size of its public key remains a practical limitation. In this work, we propose a McEliece-type cryptosystem using large minimum distance error-correcting codes derived from self-dual codes. To the best of our knowledge, such codes have not been implemented in a code-based cryptosystem until now. Moreover, we modify the decryption step of the system by introducing a decryption algorithm based on two private keys. We determine the parameters of binary codes with large minimum distance, which, if implemented into a McEliece-type cryptosystem, would provide a security level respectively of 80, 128, and 256 bits. For the 80-bit security case, we construct a large minimum distance self-dual code of length 1064, and use it to derive a random punctured code to be used in the corresponding McEliece-type cryptosystem. Compared to the original McEliece cryptosystem, the key size is reduced by about 38.5%, although an optimal decoding set is yet to be constructed to make the new system fully defined and usable.