How long can the Catholic Church be without a pope?
There must always be a perpetual line of successors of St. Peter, validly elected by the members of the college of cardinals.
One can speculate on many things and this question is also one of them.
Let us start with the basics before going on into a little more detail about this subject.
In Pope St. John Paul II's Apostolic Constitution, Universi Dominici Gregis (February 22, 1996) we read who exactly can vote in a papal conclave.
THE ELECTORS OF THE ROMAN PONTIFF
The right to elect the Roman Pontiff belongs exclusively to the Cardinals of Holy Roman Church, with the exception of those who have reached their eightieth birthday before the day of the Roman Pontiff's death or the day when the Apostolic See becomes vacant. The maximum number of Cardinal electors must not exceed one hundred and twenty. The right of active election by any other ecclesiastical dignitary or the intervention of any lay power of whatsoever grade or order is absolutely excluded.
THE ELECTION PROCEDURE
Since the forms of election known as per acclamationem seu inspirationem and per compromissum are abolished, the form of electing the Roman Pontiff shall henceforth be per scrutinium alone.
I therefore decree that for the valid election of the Roman Pontiff two thirds of the votes are required, calculated on the basis of the total number of electors present.
Should it be impossible to divide the number of Cardinals present into three equal parts, for the validity of the election of the Supreme Pontiff one additional vote is required.
Pope Benedict XVI added a few changes to the election process before resigning his office as Supreme Pontiff in his Moto Propria, De Aliquibus Mutationibus In Normis De Electione Romani Pontificis]2 of June 7, 2007.
Pope Benedict XVI has changed the rules that will decide his successor after his death, returning to a traditional method requiring a two-thirds majority. - Pope alters voting for successor
As long as there enough cardinal electors alive to elect a pope and providing that Papal Conclave rules are not altered by legitimate authority, the greatest length of time the Catholic Church can remain without a Supreme Pastor would be (taking into account the age of the youngest cardinals) 15-30 years (perhaps a little more, but it is only a guess), depending on the variables at that time. How long it takes for Cardinal Electors to die off is a variable we can only guess at!
I cannot find any Vatican document stating a minimum number of cardinals to make a papal election valid, but there are some who think that this number could be reduced to 4. But again I cannot find any further documentation, although years ago, I read that this as being the fewest number of cardinals necessary to vote for a pope.
Cardinals that reach the age of 80 before the day the Holy See becomes vacant may not vote in a conclave. If they turn 80 after Sede Vacante has already commenced, they are still eligible to vote!
A list of Cardinal Electors may be found here.
As long as there are 4 cardinal electors an election can be held!!!?