You Wrote: A gentile, even an idol-worshipper, could volunteer a sacrifice to be offered at the Temple (Menachot 73b). The only sacrifice accepted from him was a burnt-offering (Olah); Jews, on the other hand, whether they were men or women or converts to Judaism could volunteer any kind of sacrifice (Hilchot Maaseh Hakorbanot 3,2). There are ten levels of sanctity in the Land of Israel, eight of them within the Temple Mount. As the level of sanctity ascends, those permitted to enter decrease. A gentile, for example, was not permitted to enter even the “Chail”- which was on the outer perimeter of the “Ezrat Nashim” (women’s court). Jews who were not Cohanim could enter the “Ezrat Cohanim” (the court of the Cohanim) even if they were free of all defilement (“Tumah”), only to play their role in the offering of their sacrifice (Hilchot Beit HaBechira 7,19). Q: What do we do now that there is no sacrifice for atonement? How do we deal with atonement?
Atonement today is achieved through repentance- mending any damage caused, regret over the past, acceptance of a new standard of behavior for the future- and prayer including turning to G-d and asking for forgiveness for the sins committed.