Glenn Morton has on his web-site an article Early Church Fathers on Genesis by John Tobin. The point in this article is that not all the early church fathers didn't believe in a six days creation and a young earth. The article does not deny that some of the ECFs believed in a six times 24 hours creation.
Interestingly, a YEC website, www.creationism.org, has an article named The Early Church Fathers Believed in A Young Earth & Recent Creation, which claims that Origen is the only ECF that maybe interpreted the days of creation as anything but 24 hours days.
And the article begins with a quote from The Epistle of Barnabas:
The Sabbath is mentioned at the beginning of the creation: "And God made in six days the works of His hands, and made an end on the seventh day, and rested on it, and sanctified it." Attend, my children, to the meaning of this expression, "He finished in six days." This implieth that the Lord will finish all things in six thousand years, for a day is with Him a thousand years. And He Himself testifieth, saying, "Behold, to-day will be as a thousand years." Therefore, my children, in six days, that is, in six thousand years, all things will be finished.
The article supplies a few more quotes of similar content.
So, not only did some of the ECFs believe in a literal six days creation, some of them also believed that everything would be finished in 6,000 years; that is, the end of the world would come 6,000 years after the start of the creation.
This is even clearer in a quote given a few paragraphs later. In Against Heresies, Book 5, Irenaeus writes:
For in as many days as this world was made, in so many thousand years shall it be concluded. And for this reason the Scripture says: "Thus the heaven and the earth were finished, and all their adornment. And God brought to a conclusion upon the sixth day the works that He had made; and God rested upon the seventh day from all His works." This is an account of the things formerly created, as also it is a prophecy of what is to come. For the day of the Lord is as a thousand years; and in six days created things were completed: it is evident, therefore, that they will come to an end at the sixth thousand year.
So, origins may for YECs simply be a part of eschatology, and that might explain why it is so important for them that the earth not be much older than 6,000 years old. As I have mentioned elsewhere, Gerald Aardsma, who formerly worked with ICR, had to leave, because he accepted to push the upper limit to 12,000 years, so it's not just a question of Genesis 1; there is clearly more at stake.
And that more might well be that the real question for the YECs is not, how long the earth has existed, but how long it will continue to exist.
However, is this idea supported by the Bible? The Bible does operate with longer cycles that are based on shorter cycles, such as the sabbathical cycle of seven years, a week of years, based on the seven days week, and the jubilar cycle, fourtynine years made up of seven sabbathical cycles. But these are cycles, something repeating itself, not something with a final ending. So, this particular YEC idea of the earth lasting for as many thousand years as the creation in Genesis 1 spanned days would seem not to be supported by the Bible.