How can 100Base-T achieve 100 meter distance at 125Mhz clock?
From your question:

the signal should only be able to travel about 280,000,000 * (1/125,000,000)
 =~2.24 meters before the next clock cycle starts.


From one of your comments:

"I thought since the signal is essentially a difference in voltage, when you
change the voltage, the signal that's midway across the wire gets clobbered?"


You have provided the answer yourself: the next bit that is put on the line won't be able to clobber the bit that is halfway because that next bit can travel only at a fixed speed, and the bit that is halfway travels at the same speed. They will never meet (except at the far end, if it not properly terminated, but that's a different story).

If you want to visualise this, take a long rope in one hand , the other end of the rope fixed. Now you can wave your end of the rope to cause a wave in the rope to travel from your hand to the other end. For a long rope, there can be miltiple waves all traveling towards the other (fixed) end, without overtaling each other.