> Clearly you don't have two angles to do this

For nearby stars you do. You measure their position in the sky, and then you measure again 6 months later when Earth is on the opposite side of the Sun. Twice the Sun/Earth distance is a short baseline compared to the distance to stars but angle measurements are precise. Stars move relative to the Sun so you need at least three measurements, and in practice you try to get even more to reduce uncertainties.

That method works up to ~10,000 light years or so (with a somewhat lower precision for distances beyond that). For stars farther away you use the cosmic distance ladder, which uses stars with well-known behavior nearby to determine the distance of equivalent stars farther away. Objects next to these can then be used to estimate the distance of even farther objects with the same method.