Are Diesel engines constructed differently, so that they have a higher stroke volume? (longer stroke, smaller combustion chamber etc.)
Is it because they are designed different to have a greater volume?
Let's ask my question in a different way: What do you need to change of the parts in the engine to create a certain compression ratio? Which constructional factors do have influence on the compression ratio?
As the name implies, the compression ratio is a ratio between volumes of the chamber between bottom dead centre and top dead centre. Increasing the ratio requires changes that are quite fundamental: longer stroke and/or smaller combustion chamber. Another caveat is that in diesels forces exerted on the piston, its rod, crankshaft and shaft bearings are higher so those components are heavier and sturdier. For this reasons, diesels and gasoline engines are designed separately, share no basic components and it's pretty much impossible to convert one to another.
The most common method of increasing compression ratio on existing engine is to skim the head - that is "lower the ceiling" in combustion chamber. This unfortunately makes everything mounted in the head (spark plug and possibly valves) closer to piston when it's in top position, so you have very limited room for skimming before piston starts touching things it's not supposed to touch.
Another problem is that shapes of combustion chambers in modern engines are very carefully designed to achieve good flame propagation and even combustion. So while in theory one could design a simple chamber insert just to occupy some volume and therefore increase the compression ratio, it would totally mess up the combustion (not even mentioning the possibility of it getting lose and wrecking the engine).
After all those "why it's impossible" reasons, here's one thing that can and actually is done: Because compression starts only after all the valves are closed, you can lower the compression by delaying closure of the intake valve. So, in first part of the "compression" stroke, air is expelled back into intake manifold (so no compression actually takes place), then the valve closes and only the remaining part of the stroke actually compresses. This trick is used in so called "Atkinson cycle" engines that achieve better fuel efficiency at cost of lower power, eg in Prius. (The extra efficiency comes from power stroke being longer than compression stroke, not from lowering compression ratio.)
Bottom line: by changing intake valve timing you could lower the compression ratio almost all the way down to 1:1.