Remember that a timer is just a counter. So it isn't (relatively) difficult to design and manufacturer a counter that runs at gigahertz speeds.
Also, they don't have to be digital. You could charge a precision capacitor and measure the voltage, as another example. (this is how high speed pulse triggers work in modern digital oscilloscopes.)
You could potentially interleave counters to get a higher effective resolution, again, a technique used by modern digital oscilloscopes. (they have digitizers that run up to 100Gigasamples/s, which are many digitizers interleaved).
I would also imagine these devices are taking many, many readings and averaging to account for some drift in the components used.
Don't get wrapped up in the numbers. The techniques used are simple, just at a very fast speed...
If a timer is that fast then we could use a coil of fibre optic cables as a form of memory storage by measuring the cable and finding the smallest pulse size capable with the clock and then fill up fibre
You could, but that would be an expensive form of memory. If you think of dram, it works in a similar way. Charge up a cap, measure the voltage. Periodic refreshing is needed. Same idea you have, but with light.