Lord Shiva says
A shudra who is pure in his deeds, pure in his soul and has conquered his senses can obtain the status of becoming a brahmana. Brahma himself said this. It is my view that a shudra who is naturally pure in his deeds should be known as one who is purer than a brahmana. Birth, sacraments, learning and humility are not sufficient reasons for a person to be a brahmana. Conduct is the only reason. All the brahmanas in the world obtain that status because of conduct. O one with the beautiful hips! A person who possesses that good conduct obtains the status of a brahmana.
- 131, Dana Dharma parva, Mahabharata.
According to Yudhishtira any person who has the qualities of truthfulness, charity, lack of anger, lack of cruelty, nonviolence and compassion is a Brahmana and any person without these qualities are Shudras
The serpent asked, “Truthfulness and knowledge of the brahman can be found in all the four varnas. O Yudhishthira! Truthfulness, charity, lack of anger, lack of cruelty, nonviolence and compassion can also be found among shudras. O lord of men! You have said that the object of knowledge is beyond happiness and unhappiness. But there is nothing that is free from either. I do not think such a thing exists.”
Yudhishthira replied, “If these traits, not even found in a brahmana, are seen in a shudra, he is not a shudra. A brahmana in whom a brahmana’s traits are not found, is a shudra. O serpent! It is said that one in whom these traits are found is a brahmana. O serpent! One in whom these traits are not found, is marked as a shudra.
- 177, Ajagara parva, Mahabharata.
It's further explained in the text
Any restraint in anything then becomes unnecessary. O brahmana! One may be born as a shudra. But if he is established in his good qualities, he will become a vaishya, and even a kshatriya. Similarly, one who follows worthy conduct can become a brahmana.
- 203, Markandeya samasya parva, Mahabharata.