What is the meaning of "freeing a slave" in our current era, and how can one atone for the sin of killing a soul if it's impossible to reach the family of the deceased or to fast for two consecutive months? And is giving charity in money to the poor sufficient as an expiation for that, and what is its amount?

There is nothing that stands between you and repentance. Allah is vast in forgiveness, and He does not burden a soul beyond its capacity. Therefore, exert what you are able to. If your inability to fast for two months is temporary, then postpone it until a time when you are able to fast. If it is permanent, then there is nothing upon you according to the majority of scholars. It was also said: You must feed sixty poor people. As for the blood money (diyah), it is upon the 'Aqilah (male relatives on the father's side) and is not waived by statute of limitations. If reaching the family of the deceased is impossible, then this money is treated like a debt in one's responsibility, and its value can be given as charity on behalf of its rightful owners. And Allah knows best.