On what basis does the Catholic Church teach that women cannot be ordained?

Whether or not one agrees with the interpretation, there is a reasonable biblical case to be made that a woman should not preach or teach in the church. Paul writes in 1 Tim 2:

8 I desire then that in every place the men should pray, lifting holy hands without anger or quarreling; 9 likewise also that women should adorn themselves in respectable apparel, with modesty and self-control, not with braided hair and gold or pearls or costly attire, 10 but with what is proper for women who profess godliness—with good works. 11 Let a woman learn quietly with all submissiveness. 12 I do not permit a woman to teach or to exercise authority over a man; rather, she is to remain quiet. 13 For Adam was formed first, then Eve; 14 and Adam was not deceived, but the woman was deceived and became a transgressor. 15 Yet she will be saved through childbearing—if they continue in faith and love and holiness, with self-control.

A theological justification for the point then is this - Eve was deceived, and therefore all women can no longer be trusted. It is, in fact, Scriptural, using a very plain sense reading of the text.

This was precisely the view of John Chrysostom (the Golden Mouth), who in full 4th century fair writes:

“But I suffer not a woman to teach.” “I do not suffer,” he says. What place has this command here? The fittest. He was speaking of quietness, of propriety, of modesty, so having said that he wished them not to speak in the church, to cut off all occasion of conversation, he says, let them not teach, but occupy the station of learners. For thus they will show submission by their silence. For the sex is naturally somewhat talkative: and for this reason he restrains them on all sides. “For Adam,” says he, “was first formed, then Eve. And Adam was not deceived, but the woman being deceived was in the transgression.” If it be asked, what has this to do with women of the present day? it shows that the male sex enjoyed the higher honor. Man was first formed; and elsewhere he shows their superiority. “Neither was the man created for the woman, but the woman for the man.” (1 Cor. 11:9.) Why then does he say this? He wishes the man to have the preëminence in every way; both for the reason given above, he means, let him have precedence, and on account of what occurred afterwards. For the woman taught the man once, and made him guilty of disobedience, and wrought our ruin. Therefore because she made a bad use of her power over the man, or rather her equality with him, God made her subject to her husband. “Thy desire shall be to thy husband.” (Gen. 3:16.) This had not been said to her before.

John Chrysostom. (1889). Homilies of St. John Chrysostom, Archbishop of Constantinople, on the First Epistle of St. Paul the Apostle to Timothy. (J. Tweed & P. Schaff, Trans.)A Select Library of the Nicene and Post-Nicene Fathers of the Christian Church, First Series, Volume XIII: Saint Chrysostom: Homilies on Galatians, Ephesians, Philippians, Colossians, Thessalonians, Timothy, Titus, and Philemon (p. 435). New York: Christian Literature Company.

And, this is not just 4th Century prejudice. No less than the NIGTC (The New International Greek Testament Commentary) says of this:

Some have suggested that Paul conveys here only a note of personal disinclination (cf. Phillips’s translation: “Personally, I don’t allow”). But such a suggestion misunderstands the authoritativeness of ἐπιτρέπω when used by Paul (cf. Robertson: “Paul speaks authoritatively”), which is demonstrated by a close analysis of the three occurrences in Paul (1 Cor. 14:34, a parallel; 16:7, an action of the Lord; here). The strength of the prohibition here is underlined by Paul’s appeal to the creation order (v. 13, γάρ); in 1 Cor. 14:34 the prohibition is correlated to “the law” (undoubtedly the same OT teaching as here in v. 13) and is further delineated by his covering statement in v. 37, “the things that I write to you are the Lord’s commandment.”

That said, there are varying interpretations and contra-indications to such a plain instruction. (2 John being addressed to "the elect lady and her church" comes to mind.) But, if the standard interpretation is the sensus plenior here, then the Bishop of Rome would simply being saying that the church has no authority to create a new teaching out of a very old Scripture. After all, the order of things was established far before him, and woe betide the one who tries to change it.

When Pope John Paul II definitively reaffirmed the teaching in 1994 he referred back to the doctrinal document Inter Insigniores of 1976, which, he says, "recalls and explains the fundamental reasons for this teaching". Without attempting to go through Inter Insigniores in depth here (it's not really that long to begin with), here are some quick excerpts that summarize the main points it covers:

1. Constant and universal tradition: "up to our own time ... the practice has enjoyed peaceful and universal acceptance. ... The Church's tradition in the matter has thus been so firm in the course of the centuries that the Magisterium has not felt the need to intervene in order to formulate a principle which was not attacked, or to defend a law which was not challenged. ... The same tradition has been faithfully safeguarded by the Churches of the East"
2. Practice of Christ himself: "Jesus Christ did not call any woman to become part of the Twelve"
3. Practice of the Apostles: "The apostolic community remained faithful to the attitude of Jesus towards women"
4. These practices not a sign of conformity to the social mores of antiquity: "an examination of the Gospels shows ... that Jesus broke with the prejudices of his time, by widely contravening the discriminations practised with regard to women. ... [S]ocial and cultural conditioning did not hold back the Apostles working in the Greek milieu, where the same forms of discrimination did not exist. ... there is no reason for accusing [St. Paul] of prejudices against women, when we note the trust that he shows towards them and the collaboration that he asks of them in his apostolate"
5. Clarity of the sacramental sign: "when Christ's role in the Eucharist is to be expressed sacramentally, there would not be this 'natural resemblance' which must exist between Christ and his minister if the role of Christ were not taken by a man"
6. Ministry in the Church a vocation, not a right: "to consider the ministerial priesthood as a human right would be to misjudge its nature completely: baptism does not confer any personal title to public ministry in the Church. The priesthood is not conferred for the honour or advantage of the recipient, but for the service of God and the Church; it is the object of a specific and totally gratuitous vocation: 'You did not choose me, no, I chose you, and I commissioned you' (Jn 15:16; cf. Heb 5:4)"
7. Vocation to public ministry not purely subjective: "Since the priesthood is a particular ministry of which the Church has received the charge and the control, authentication by the Church is indispensable here and is a constitutive part of the vocation: Christ chose 'those he wanted' (Mk:13)"

Finally, John Paul II seemed keen to put the notion to rest that there was some sort of unjust discrimination happening here:

Furthermore, the fact that the Blessed Virgin Mary, Mother of God and Mother of the Church, received neither the mission proper to the Apostles nor the ministerial priesthood clearly shows that the non-admission of women to priestly ordination cannot mean that women are of lesser dignity, nor can it be construed as discrimination against them. Rather, it is to be seen as the faithful observance of a plan to be ascribed to the wisdom of the Lord of the universe. (Ordination Sacerdotalis 3)