OUT OF THE BALLPARK
1. At each successive moment the number of balls keeps doubling: 1, 2, 4, 8, and so on. So at midnight the number of baseballs will be the limit of this series, namely infinity.
2. Each time I throw balls into the room, the person inside the room throws half as many back out. I threw in an infinite number of baseballs, so the person inside threw out half of infinity, leaving half of infinity inside. (But of course half of infinity is still infinity.)
3. Every ball thrown in is eventually thrown out before midnight, so no balls remain.
4. The question itself is flawed because there is a limit to the speed at which balls can be thrown. (And, in theory, if you keep splitting the remaining time in half, some would argue that midnight would never strike.)