You have four choices.
All of these choices require you drive local grounding rods and also have a ground wire running back to the main building, and tie both of those together. You cannot do one as a substitute for the other, they do different things.
Take that wire back and get /3
/3 cable will let you hook it up in the normal way you expect.
If the people who sold you this wire were the same ones who told you you didn't need /3 cable, they ought to take it back!
Supply 120V only
Bare goes to ground. White goes to neutral. Black goes to hot.
You can carry 40A@120V over that cable. That will happen at about 6% voltage drop (114V) which is a little concerning but I wouldn't worry too much. 20A will happen at 3% drop.
40A is not very useful for receptacles, so you'll need to bust that out into a subpanel. Get a large subpanel (for future expansion) and feed to one hot bus. Only every other row of breakers will light up.
Supply 240V only
Voltage drop is less than 3% at 240V. Here, ground is ground. Tape the last few inches of the white a color like black or red and it is hot 1. Black is hot 2. You do not have neutral and cannot power any 120V load. You will have a subpanel with no neutral bus. It will have a ground bus.
This can work if the primary load is a pump, hot tub, pool, well. etc. 240V lighting is readily available. For small amounts of 120V power, like a 15A receptacle circuit, do not use a step-down transformer, use a "mini" separately derived service, with a smaller (more affordable) transformer.
The problem with this setup is it is indistinguishable from a normal main panel where ground/neutral are the same bus. Of course people will want 120V loads out there, and less skilled electricians will blithely hook it up (and it'll work by bootlegging). A skilled electrician will go "You forgot neutral" and call the installation a botch job - no, it's a 240V only panel.
Separately derived service
Ground doesn't matter. White is taped with a color. Black and white both go to the 240V primary of a transformer. The secondary gives 120/240V split phase. Feed that into a service panel which is a main panel owing to the isolation caused by the transformer. In this main panel, bond neutral to your local ground rod.
Since electricity is transmitted at 240V, voltage drop would be less than 3%, and would remain so after being transformed down to 120V.
For the full 40A@240V, that's 9600 VA so a 10KVA transformer would be needed. Those are several hundred dollars at least, so it's probably cheaper just to use the right cable. However if most of your loads can work on 240V and you only needed 120V for a receptacle string or two, you could use a much smaller 1.5KVA or 5 KVA transformer, which are cheaper and readily available used.
Honestly, this plan works better over longer distances, where you can go even further by using 2 transformers back to back to step up transmission voltages.