All materials have structural limits. A column, made of a certain material in a certain gravitational field, will either crush or buckle under its own weight if it is too high. This still applies if the gravitational field is the self-gravity of the structure itself.

Dynamic support might be used. One concept is you make a sort of long skinny particle accelerator and the particles going back and forth exert a reaction force on the endpoints supporting the far end. But even with that, the bottom end loop has to be able to support the load.

Based on known materials, even a solid Dyson sphere is impossible without dynamic support. A realistic Dyson sphere would instead be simply a swarm of satellites so dense it can absorb nearly all its star's light.

The ultimate limit is that the volume of a region of space scales with radius cubed, but the critical mass above which a black hole is formed scales linearly with radius. Thus the larger a structure is, the less dense it must be to stay below that critical mass. (Which in any case could only be approached by transporting matter in from other galaxies, since the starting galaxy was obviously not a black hole.)

Once you're at the point that the "structure" is a thin mesh of wires or tubes stretching over interstellar distances, well, why have the structure at all instead of just a bunch of free-floating spacecraft that can share power and data with lasers or something?