Suppose tug-of-war Team 1 and Team 2 both had of an equal number of tuggers, all of which are equally good at tug-of-war. If everyone is pulling for their respective team, the rope won't budge. Yet, for any such tug-of-war, a team which loses all its odd-numbered members would surely and suddenly be at a disadvantage. But not if both teams had an infinite number of members! If you think actual infinities are possible, then despite all our intuitive and inductive real-world evidence, you'd have to say Team 2 would not suffer any disadvantage whatsoever.1 In fact, instead of imagining only every other person holding on, we can imagine only every hundredth, or trillionth, individual holding on for Team 2. Everyone else on Team 2 suddenly lets go and joins the other side. The rope doesn't budge! In fact, suppose the remaining members on Team 2 now suddenly shift to pulling at only 1% of their strength. Despite this, still nothing changes. Team 1 never starts winning.
Suppose a hotel existed which contained an infinity of occupied rooms.2 If the rooms are all occupied, then there are no vacancies and so clearly no room for newly arriving guests. If infinities are possible, however, a new guest could be accommodated. He could be given room 1 by moving the guest in room 1 to room 2, and the guest in room 2 to room 3 and so on). In fact, an infinity of new guests could be accommodated, simply by moving every person in a room n to room 2n (e.g. room 1 to room 2, room 2 to room 4...), freeing up all the odd numbered rooms. In fact, even an infinity of coaches with an infinity of members per coach could be accommodated.
Imagine a 4-color rainbow popsicle (or a sufficiently gunky replicate of it) whose segments repeat red, yellow, blue, violet. It can simultaneously have an infinity of stacking segments and still be just 4 inches tall in the following algorithmic way: Holding it upright, start with a 2 inch red segment on the bottom, and let every following segment moving upward be half the height of the segment below it, ad infinitum. If actual infinities are metaphysically possible, there is no highest color (since there is no last member of an infinity). But that's absurd. Just hold the popsicle below your chin and look down.1
It is metaphysically possible for an infinity (ℵ0) to be instantiated in the world. This is relevant because if they are possible, then they trivially do not entail the possibility of metaphysically impossible scenarios.