Jesus defined eternal life in John 17:3. It is knowing (in an intimate sense) the Father and the Son.
Therefore, eternal death is being separated from the Father and the Son. This would have no meaning if death meant unconsciousness, but it doesn't. It means separation. Physical death is separation from the body and spiritual death is separation from God.
In Romans 7, when Paul says that the sin came to life and he died, he is clearly not referring to physical death, but rather to spiritual death -- separation from God.
Is there then any time when a person is not separated from God if he enters eternal death? Does he then become unconscious? If so, Jesus' words in Matthew 25 lose their meaning.