It is a common belief that sudras can"t learn vedas from Hindu scriptures but according to vedas itself a shudra can learn vedas-
Sukla Yajurveda (XXVI. 2). यथेमां वाचं कल्याणीमावदानि जनेभ्यःब्रह्मराजन्याभ्यां शूद्राय चार्याय च स्वाय चारणाय॥ Meaning-
The way I gave this knowledge of Vedas for benefit of all humans, similarly you all also propagate the same for benefit of Brahmins, Kshatriyas, Shudras, Vaishyas, Women and even most downtrodden. The scholars and the wealthy people should ensure that they not deviate from this message of mine."
Here is how Ralph T Griffith translates this verse-
That I to all the people may address this salutary speech,To priest and nobleman, Sûdra and Arya, to one of our own kin and to the stranger Dear may I be to Gods and guerdon-giver. Fulfilled be this my hope: be that my portion!
Here is what swami vivekananda says about this verse:
Thus says the Shukla Yajur Veda (XXVI. 2). Can you show any authority from this Veda of ours that everyone has not the right to it? The Purânas, no doubt, say that a certain caste has the right to such and such a recension of the Vedas, or a certain caste has no right to study them, or that this portion of the Vedas is for the Satya Yuga and that portion is for the Kali Yuga. But, mark you, the Veda does not say so; it is only your Puranas that do so. But can the servant dictate to the master? The Smritis, Puranas, Tantras — all these are acceptable only so far as they agree with the Vedas; and wherever they are contradictory, they are to be rejected as unreliable. But nowadays we have put the Puranas on even a higher pedestal than the Vedas! The study of the Vedas has almost disappeared from Bengal. How I wish that day will soon come when in every home the Veda will be worshipped together with Shâlagrâma, the household Deity, when the young, the old, and the women will inaugurate the worship of the Veda!( Quoting from the complete works of swami vivekananda volume 3 http://www.ramakrishnavivekananda.info/vivekananda/volume_3/lectures_from_colombo_to_almora/the_religion_we_are_born_in.htm)
So if according to Veda a shudra can learn vedas,then why do puranas say that sudras are not allowed to learn vedas?