We have to read the verse from the perspective of the speaker and his audience. To whom was Jacob / Israel speaking? If we honestly answer that question, then we can know the meaning of "the last days" as God used them in scripture.
Jacob was telling each of his sons what would befall THEM in THEIR last days. This is the first use of the "last days" or "latter days" (YLT) in the scriptures. It concerned the last days of each of the tribes of Israel.
The last days were always about the end of days of the children of Israel, and the Mosaic covenant. Notice specifically what Jacob told Judah in verse 10.
"The sceptre shall not depart from Judah, nor a lawgiver from between his feet, until Shiloh come; and unto him shall the gathering of the people be." (KJV)
"The sceptre turneth not aside from Judah, And a lawgiver from between his feet, Till his Seed come; And his [is] the obedience of peoples." (YLT)
The promised seed of Gen. 3:15 was Christ. The scepter would remain with Judah until Christ came. And, unto Christ is the gathering / obedience of all people.
Christ came and was manifested in the first century AD. The sceptre passed from Judah to Christ - the dominion of the kingdom passed from fleshly Israel to spiritual Israel - at the cross of Christ.
Judah was the last of the tribes to remain until Christ came. God maintained the land of Judah for Christ's manifestation (Ezek. 37:19-25). The last days were always speaking of the old covenant between God and fleshly Israel. The Bible was always leading the people to Christ.
Once the covenant was fulfilled in Chirst (Matt. 5:17; Rom. 8:4), there was no longer any need for that fleshly sacrificial system at that temple in Jerusalem. Christ became the sacrifice for all.
"16 `This [is] the covenant that I will make with them after those days, saith the Lord, giving My laws on their hearts, and upon their minds I will write them,'
17 and `their sins and their lawlessness I will remember no more;'
18 and where forgiveness of these [is], there is no more offering for sin." (YLT)
That was the promise prophesied from Jer. 31: 33-34. Under the new covenant of the gospel of Christ we have forgiveness in Him. His word is written on our hearts, not on tablets of stone. We have forgiveness in Christ, as it is His blood that covers us when we are baptized into Christ. That means there is NO MORE SIN OFFERING at an earthly temple that is going to forgive our sins.
The last days were the end of the old animal sin offering system under the old covenant. They were not and never were speaking of the "end of the world".
Further reading at my blog:
"Frequent Mistakes - Part VI: The End of The World, or ???" at ShreddingTheVeil
"Frequent Mistakes - Part III: The Last Day" at ShreddingTheVeil
"The Signs of Revelation - Part I: The Time of His Coming" at ShreddingTheVeil
"The Valley of Dry Bones: A Picture For The Children of God" at ShreddingTheVeil