How to determine the structure of organic molecules without spectroscopy
One thing that was well known, even in the early days, was elemental analysis. It is Justus von Liebig’s contribution to have worked out the sum formulae of hundreds of natural compounds. This generally worked by combusting a given sample of a compound, capturing the vapours, absorbing the water with desiccants such as calcium chloride, absorbing carbon dioxide by potassium hydroxide (forming a carbonate), weighing the different parts and thereby calculating the mass of carbon and hydrogen in a given sample. Tests for nitrogen and sulfur also existed, but I can’t remember them off the top of my head. Oxygen content was indirectly calculated as being the remainder.

Chemists since the advent of their science had done everything to generate pure compounds. It was widely accepted even then that a pure compound was one with a defined melting or boiling point, and liquids could also be analysed by their refractive index and density. This provided a rather good library of methods for realising ‘is my compound pure?’, and ‘is it the same as $\ce{X}$?’

Another very old concept was that of valencies. By analysing samples of known and simple elemental composition, it was realised that certain elements bonded to a certain number of other elements, e.g. that a hydrogen would always require a bonding partner of value 1. Somebody (I can’t remember who but I could look it up in my lecture notes) proposed a valency of 4 for carbon. Together with a valency of 2 for oxygen, you had most pieces of the puzzle in hand.

It was recognised by Liebig and Wöhler, that isomers existed, i.e. compounds of the same elemental composition but distinct. (This is also a consequence of the ammonium cyanate experiment Wöhler performed to create urea, breaking the wall between organic and inorganic chemistry and proving vis vitalis to be inexistent — both are isomers of each other.) So there must have been something important in the structure of molecules that needed to be discovered.

A lot of further clues then came from reactions that transformed known compounds into other known compounds. For example, oxidation would transform some known compound $\ce{C3H8O}$ into acetone $\ce{C3H6O}$. This also lead to Liebig’s theory of radicals — where radicals is something a modern chemist would refer to as functional group. With these clues and radicals, the task of putting the puzzle together to pinpoint certain structures became solveable.

From then on, it all depended on whether the initial assignments were correct (they mostly were). You would go ahead either breaking down unknown compounds by known reactions after having established their functional groups or resynthesising them from known fragments in other known reactions to arrive at a final product.

Working out the difference between hexan-2-ol and hexan-3-ol would have been a none-trivial task; that is true. One variant seems to be starting from hex-2-ene and adding water to the double bond. You would get two different products and with the knowledge of structure in hand, you would realise that they had to be isomers of each other and had to be the 2- and 3-isomer respectively. How did you get hex-2-ene? Oh, maybe you managed to dehydrate hexan-2-ol into two different products, hex-1-ene and hex-2-ene. How did you get hexan-2-ol in the first place, then? Addition to hex-1-ene sounds reasonable. How to get that? Dehydration of hexan-1-ol. That final one was easily established as a special hexanol, since it could be oxidised to hexanoic acid, not only to a ketone. It is all possible but requires a lot of work. Think how lucky we are that we can record an NMR spectra within minutes (or a complete set over night) and solve the structure the next day!

It is entirely possible that for the cholesterol family, the exact ring connections were initially not known, only that there is some type of a fused ring system that can be modified in certain positions (not all!). It would have taken extra time to break down the ring system accordingly. And Emil Fischer’s work on carbohydrates probably warrants one or two questions of its own.

And then, crystallography happened, and all of a sudden was it possible to see structures. It must have seemed like the dawn of a new era.