If we compare Luke 13:25 with Matthew 10:25, we should be able to see that Jesus is the master of the house in the parable, and that it is he who closes the door after he is risen up (Luke 13:25). So, understanding what Jesus refuses to do for the many is necessary to understanding the meaning of saved in John 10:9 and Luke 13:23, because certainly the rabbi who asked Jesus the question didn’t believe he needed to be saved from his sins. He believed this was provided for through the Temple sacrifices. Moreover, sin is never mentioned by Jesus or anyone else in either Luke 13 or John 10. So, what does Jesus mean by saved in John 10:9, and is he speaking of the same thing that the rabbi has in mind in Luke 13:23?
Notice what Jesus concludes in the parable. The master of the house closes the door, after which those outside are not saved (Luke 13:25; cf. John 10:9)? What Jesus means to say is that he offered the Kingdom of God to the nation of Israel during his public ministry. Afterward, he offered the same Kingdom to the nation through his disciples for approximately 40 years, but the Jewish nation repeatedly refused to allow Jesus to rule over them (cf. Luke 19:27), even going to the extent of persecuting those Jesus sent to them (cf. Matthew 23:34-36).
However, once Jesus received the Kingdom (Luke 19:12, 15), he rose up and closed the door (Luke 13:25), so that the nation was no longer offered the Kingdom, and immediately afterward Jesus judged them for rejecting him (Luke 19:27; cf. Matthew 26:63-64; Mark 14:61-62). The war between the Jews and Rome broke out quite suddenly in 66 AD. From that point on, the door was closed, because Jesus had risen up in order to judge the nation for rejecting him as their Messiah and King.
In Jesus’ parable the many come and knock at the door (G2374) after it is shut (Luke 13:25). Matthew refers to this as the time when the Kingdom of Heaven comes (Matthew 7:21-22) and at the time of the coming of the bridegroom, who is Jesus, the Messiah (Matthew 25:6, 11-12). So, they are requesting that the Messiah would act on their behalf, which is the context of the time of the Jewish war with Rome. The nation had hoped that the Messiah would arise out of that war and free them from the yoke of Rome.
In Luke 13:27 Jesus claimed he didn’t know (eido – G1492) these people who knocked or sought him to act on their behalf. That is, he didn’t know them as his disciples. During Jesus’ public ministry, these same people sought to kill him. They didn’t seek his help at that time. Neither did they repent afterward and submit to his authority and trust in him as their Messiah. Rather, they persecuted and killed those servants he sent to them. He never knew them to trust in him or regard him as their Lord before the time he rose up to judge his people.
Therefore, in the sense that the Kingdom was no longer offered to the Jewish nation, the door was then and continues to be shut, but in the sense that the Kingdom of God is offered to whomsoever will come on an individual basis (Revelation 22:17) and submit to Jesus as Savior, the door is always open.