I found some actual step-recovery diodes at Mouser. They're not the fastest ones available, with only a 150 ps maximum transition time, but that's still pretty good. They're available in SOT-23 and SC-79 packages, so I bought a few in each. I was able to test one today with the same setup I had used to test a 1N914 earlier.

Here's the input pulse (blue) vs the output (green). The diode has sharpened the fall time considerably, from 7.8 ns to 362 ps according to the scope auto-measurements at this time-scale. It's actually much faster than that.

The circuit to test this diode is the same one I used before:

In this case, the diode was in a SOT-23 package, so I used a little adapter PCB to mount it to two BNC connectors.

Here's the falling edge of that pulse again, at 200 ps/div. The scope now says it has a 278 ps fall time, but guess what? That's the fall-time of the scope itself! The diode probably has a fall time under the 150 ps maximum spec on the datasheet. I'll just have to find some clever way to verify it.

This seems promising. These diodes are only around $2 each, and are easily fast enough to generate strobe pulses for a 1 GHz sampling head.

I'm thinking I can drive one of these SRDs with a commercial MOSFET driver IC. The driver doesn't need to be particularly fast (a few ns transitions are fine), but needs to handle some hefty currents and have low jitter.