What sort of process would this require? Is it doable without modern technology?
I know very little about brewing, but to make a WAG, most seaweed is pretty salty. Salt, in bread at least, hinders yeast and yeast is what is used to ferment something. Yeast also works by turning sugar into alcohol. I don’t know if seaweed has any or much natural sugar.
Overall it doesn’t sound like it’s an ideal starting point. You might do better with something like cactus (if you want to go for weird and green.) http://winemaking.jackkeller.net/prickly.asp
No schooled chemist here, nor seaweed expert, but since it appears seaweeds contain carbohydrates, the task would be to convert them to sugars wherein yeast could act directly.
In the making of beer, which is the first step for many distilled beverages (mashing), certain strains of barley contain sufficient enzymatic property to convert other more recalcitrant starches, so you wouldn’t need modern technology.
Sage Rat raises an interesting point about salt impeding yeast action, likely overcome with a sufficient pitching quantity. But salt would have affect on potability.
The best way would be to get a shitload of seaweed. Sell it to someone who **wants **seaweed. Use the money they give you to buy alcohol.
Ah, thanks.
don’t ask, This is a NaNoWriMo world with very limited plant life in the region. I would not want to drink anything made from seaweed harvested just off Los Angeles 
There are tons of hits. There is quite the push to make biofuels from algea, much “seaweed” is algae.
I don’t know the exact process but I would venture that you need a good cellulase enzyme to break down the cellulose into glucose molecules. The rest is downhill, glucose with yeast to make the ethanol. Distill as normal to separate the ethanol from the rest of the stuff.
There is a lot of work going on right now to develop viable cellulase enzymes to break down cellulose (for “cellulosic” ethanol)