There are three methods for extracting oil from nuts. Cold pressing, which grinds the nuts to a paste and forces the paste through a hydraulic press, produces the lowest yield but highest quality oil. Expeller pressing uses some heat and forces the paste through an expeller using a screw. It gives a greater yield but produces a lower quality oil. Solvent extraction uses high heat and a small amount of a petroleum based solvent to extract oil. It is the most common industrial process, producing the highest yields but the lowest quality oils. Almost all soy oils are produced by the latter process.
Grind whatever nuts you plan to use into a fine paste using the mortar and pestle. The mortar is the bowl and the pestle the solid, nearly cylindrical piece you use to crush with. Push it around the bottom of the mortar in a circular motion until the nuts have become a fine paste. Do not repeatedly pound the nuts with the pestle. The near stirring motion will do the job, without losing any oils in the process.
Transfer the paste into the metal cylinder. Put the fitted insert into the cylinder after you put in the nut paste.
Put the cylinder into your shop press. Slowly apply pressure on the press until it has pressed the nut paste as far as it can.
Remove the cylinder from the hydraulic press. Place the cheesecloth over the jar. Pour the contents of the cylinder through the cheesecloth before removing the fitted insert from the cylinder.
Remove the fitted insert from the cylinder and pour the remaining contents of the cylinder onto the cheesecloth atop the jar. Let the paste sit for several hours, then squeeze the cheesecloth with all the paste in it to get the last bit of oil into the jar. Cap the jar and store after you have finished. Wipe the cylinder with a clean cloth.
The cheapest hydraulic shop presses start at $150. You can use a manual shop press, but will get a lower yield of oil. Finding a cylinder with a fitted insert may be the trickiest part. You can find them at some hobby stores and automotive supply stores. If you cannot find one there, call a supplier of hydraulic presses and describe what you want.
Do not use the cylinder for other purposes than crushing nuts, lest you adulterate the flavor of the oils you produce. Rinse it thoroughly before using it for a different type of nut. Use plain water and a cloth. Do not use soaps or detergents.