So my understanding is that once you were elected Quaestor you were eligible for Senate enrollment (assuming you were in the senatorial class). However, you always hear about there being 300 Senators, or so and so increasing or decreasing the number of Senators.
It seems to me, then, that if the Senate had a finite number you would generally have more former Quaestors than Senators. Was there a waiting list? Was a Senator's membership suspended if they were subsequently elected to Praetor or Consul? What happens when they came back from their governorship? Did they just reassume their seat?
Edit: apparently Reddit formatting doesn't work in titles. Oh well.
Sulla made membership of the senate automatic after election to the quaestorship as part of his package of reforms in 81, when he also doubled the number of senators to 600 and increased the number of quaestors to 20. Although it's likely that most ex-quaestors were enrolled in the senate in the period before Sulla's reforms, you're right that there was presumably some kind of waiting list if the senate was already fully subscribed.
Membership of the senate was for life, so politicians continued to be members throughout their careers regardless of other magistracies they held or whether they were serving as governors overseas. From some time in the late-4th century (probably between 339-318 BC, but the exact date is unclear) until Sulla's reforms, membership of the senate was managed by the censors - a senior magistracy elected every five years. The exact criteria are not preserved, but certainly included at least the following:
The censors were the only ones who could remove a senator, depending on their continued ability to meet at least the above criteria. The last one - moral behaviour and reputable business dealings - was clearly a matter of interpretation, and some censors took a stricter line than others. Cato the Elder, censor in 184 BC, has a reputation as a scourge of the senate, removing a number of senators who fell short of his incredibly exacting standards of morality, apparently including a man who had the audacity to embrace his wife in public (Plutarch, Life of Cato the Elder, 17-9).
So every five years, censors were elected and reviewed the enrolment to the senate. They expelled anyone who no longer met the qualification criteria and chose new senators to replace those who had been expelled, or had died since the last census. I should point out that this is how the system was technically supposed to work, but whether it happened exactly like this in all cases and at all times is probably unlikely.
Further reading on the republican senate:
G. P. Burton and K. Hopkins, in K. Hopkins, Death and Renewal (1983)
A. Lintott, The Constitution of the Roman Republic (1999)
T. P. Wiseman, New Men in the Roman Senate (1971).