While popes are traditionally chosen from among cardinals, this does not mean it is restricted to the cardinalate. In fact, you only have to be a baptized male to be eligible to become the next pope!
One of the most famous stories in the history of the papacy is that of Pope St. Fabian, who was chosen to become pope in the 3rd century.
Fabian was not a cardinal, bishop, or even a priest before he became the Successor to St. Peter. In fact, he was simply a farmer who came into the city to see who the next pope would be.
Farmer pope
The story is related by Eusebius in his Church History. He explains how nobody expected to select Fabian as the next pope:
They say that Fabian having come, after the death of Anteros, with others from the country, was staying at Rome, and that while there he was chosen to the office through a most wonderful manifestation of divine and heavenly grace. For when all the brethren had assembled to select by vote him who should succeed to the episcopate of the church, several renowned and honorable men were in the minds of many, but Fabian, although present, was in the mind of none.
The only reason why he was chosen pope was the descent of a dove on Fabian:
[T]hey relate that suddenly a dove flying down lighted on his head, resembling the descent of the Holy Spirit on the Savior in the form of a dove. Thereupon all the people, as if moved by one Divine Spirit, with all eagerness and unanimity cried out that he was worthy, and without delay they took him and placed him upon the episcopal seat.
Fabian received the sacrament of Holy Orders and was quickly elevated to the rank of Supreme Pontiff.
Miraculous
It certainly is a strange series of events, but one that was seen by all at the time as a miraculous intervention.
Fabian proved to be a worthy pope and died a martyr's death around the year 250.
This type of popular acclaim is unlikely to happen in the modern era, as the conclave to elect the next pope is done in absolute secrecy and behind closed doors.
However, this does not mean that the Holy Spirit will be closed out of the process. God can still guide the hearts and minds of the cardinals to choose the man he wants to be on the chair of St. Peter.
