Chaste in the context of 5:5 and ahl-al-kitab may mean: A virgin, a woman who was legally married to a man before according an other shari'a and legally separated. So in first place this would mean a widow and maybe even a divorced lady, as as far as I know divorce isn't allowed for catholic Christians this wouldn't apply for many Christians.
So ladies of this kind would be considered halal for marriage.
Non-chaste would therefore mean they have an unlawful relationship or used to have affairs.
What if the woman is not chaste? Is it possible for her to become chaste? What are the requirements for her to become chaste?
A solution for this is sincere repentance according scholars (see for example this fatwa), and i couldn't find any fatwa restricting this for Muslims only. However in this fatwa (Arabic) it is said if a non-cahste woman among the people of the book converted a Muslim is allowed to marry her:
فإذا أسلمت الكتابية ، والتزمت العفاف : جاز للمسلم أن يتزوجها ، ولو سبق منها الزنا ، ما دامت قد تابت منه ، وعزمت على تركه وعدم العودة إليه .
--- (My own translation take it with care) ---
If a woman of the poeple of the book converted, and became chaste it would be allowed for a Muslim to marry her, even if she committed zina in her earlier life, as long as she repented sincerly and intended to keep away from it and never turn back to it.
This fatwa is based on the verse (8:38) which says that Islam practically washes away former sins. This could be understood as a clear statement that repentance is the key and it doesn't matter whether the person was a Muslim or among the poeple of the book.
So this is the first step anyhow, as we on the other side are prohibited from marrying a woman who commits zina (24:3). Once she she repents sincerely and stopped committing this sin (of zina) she would be considered chaste again.
The other solution would be for her to convert as:
Islam wipes out all the previous (misdeeds) (From sahih Muslim)
And this is the answer for your former question Modern world and marriage from the people of the book so please consider deleting it!
Of course scholars always recommend you to marry a Muslim girl, so one should go ahead trying to invite her to Islam
Then, invite them to Islam, and inform them what is enjoined upon them, for, by Allah, if Allah gives guidance to somebody through you, it is better for you than possessing red camels. (From sahih al-Bukhari)
The reasons quoted for the recommendation of a Muslima for marriage are:
- the offspring being educated in an Islamic manner only when the mother is a Muslima. Else one doesn't know what will happen (this is what I often see in practice)
- most of the nowadays living woman of ahl-al-kitab have no relation to their religion and are far away from it, so they are named Christians or Jews but in reality kufar as they don't follow the rules of their shari'a, are atheists or reject religion in the whole etc..
- and back to the children, they could instead of growing up Muslims grow up Christians or Jews!