First things first: there are actually two such positions: the Crown Steward and Bailiff of the three Chiltern Hundreds and the Crown Steward and Bailiff of the Manor of Northstead. So the obvious method when two members want to resign is to appoint one to be Crown Steward and Bailiff of the Manor of Northstead and the other Crown Steward and Bailiff of the Chiltern Hundreds. These appointments are used in alternation; as the last person to resign from the House of Commons (ex-Speaker John Bercow) is now Crown Steward and Bailiff of the Manor of Northstead, the next Commons resignee will be appointed Crown Steward and Bailiff of the Chiltern Hundreds.
Each of these positions is a single position, i.e. it can be filled by one person at one time. They are appointed by, and following a request to, the Chancellor of the Exchequer. (The Chancellor may deny the application, which seems to have happened last in 1842.) One person’s appointment is immediately ended when a new person is appointed, which can be anything from minutes to years. In theory, either Crown Steward and Bailiff can apply to be released from that position, but since neither gives any salary or carries any responsibilities there seems to be no reason to.
When more than two people wish to resign at the same time, as happened, for example, following the Anglo-Irish Agreement in 1985 when all Unionist Northern Irish MP’s wished to resign in protest, the offices are appointed sequentially. As soon as an MP is appointed, they immediately cease to be a member of parliament but once they are out they are able to lose the appointment almost immediately. Thus, the lists on Wikipedia contain a big block in 1985 with a number of people appointed to both offices on the same day. The ultimate holders of these positions, William McCrea (Chiltern Hundreds) and Enoch Powell (Manor of Northstead) were both reelected in the subsequent by-elections, proving that only the appointment, not holding the position, disqualifies from sitting in the House of Commons.
As an additional amusing side fact: both offices have been held by (Irish Nationalist) Sinn Féin MP’sMPs who don’t sit in the House of Commons as they refuse to swear allegiance to the Queen. Their letters to the Chancellor with intent to resign did not explicitly mention either position and despite being appointed to one they would not admit holding an office of profit under the Crown. British legal fictions are fun!