What is the secret to baking bread with a very fine crumb?

A couple of things will help give you smaller holes:

1. Keep the hydration reasonably low (say, 60% with American-style bread flour).
2. Use some oil or butter. Try 10% (baker's ratio).
3. Knead very well, something like 10–15 minutes in a stand mixer.
4. After the first rise, normally you try to be gentle, and not press out all the air. Don't. Instead, press it out.
5. Bake in a moderate (say, 350°F) oven. You don't particularly want much oven spring here.

For even finer texture, part way through the second rise, you can press it out again, and let it start a third rise (but not to double).

Also, adding some whole wheat or rye flour will give a denser crumb. With different flavor, too, of course.

1. a small amount of fat emulsifies, trapping multiple bubbles instead of a few giant ones

2. faster overall processing: a single rise

3. denser dough and worked til highly elastic.

Dough for a sandwich loaf would be portioned into three balls and worked til taut, lined up snuggly in pan and proofed to only half of its rising capacity then baked with a lid on to further compact the crumb.

My experience is that the drier your bread dough is, the finer the crumb and smaller the holes. Try using slightly less liquid or slightly more flour.