You can reuse the wires one of several ways.
Run a separate ground wire
Since it is a retrofit, you can add ground wire. (NEC 240.130). The 3 circuits can share one ground wire if they all originate in the same panel (as of NEC 2014‘s new 240.130D). The ground wire must be installed properly, and an appropriate size for each circuit it's grounding. If your fattest conductor is 10 AWG, a 10 AWG ground wire can protect all 3 circuits under NEC 2014. If I recall, 8 AWG bare copper wire is readily available.
Hots and neutrals from the same circuit must be routed together. A 240.130 retrofit ground may follow a different route.
If an existing wire is already bare and appropriate size, you already have your ground wire!
Wire larger than 6 AWG can be redesignated as ground by wrapping it in green tape at each accessible point.
Ask your inspector to re-designate
Generally you cannot re-designate a conductor to be ground, although an entirely bare copper wire can only be a ground.
Ask your local inspector if there's a way to allow this anyway. Local jurisdictions can override Code.
Forget ground and use GFCI
GFCI's may be required anyway for your new circuits.
You can use the GFCI rules, which allow ungrounded 3-prong outlets when fed by a GFCI breaker or outlet. All the outlets must be labeled "No Equipment Ground".
They must also be labeled "GFCI protected" if they are a plain outlet fed by a GFCI upstream.
In this case, take care not to connect ground to anything. Nothing is safe except a ground wire run all the way back to the main service panel.
Make the barn a main panel, isolated by a transformer
This is a complicated one. Wire the barn as a main panel. It must have its own ground rods and neutral must be bonded to ground in the panel. However this neutral must not bond to the neutral back at the house.
It is isolated from the house by a transformer. This is a "mini" version of the service the power company provides to you.
The key is a transformer of appropriate size. It must be single-phase and with a VA rating exceeding your planned load. I often see them on Craigslist at sane prices, e.g. 5K VA for $100. Typically, you can jumper the secondary to either supply 240/120 split phase like your house, or 120-only at twice the amperage.
For amp capacity of a transformer that you've jumped for 240V, easy peasy - divide VA by 240. For instance, A 5K VA transformer is 5000/240 or 20.83 amps. It can be supplied by a 20A breaker and 12 AWG wire, and can output 20A to each side of the panel. If jumpered for 120V-only, it can provide twice that or 40A.
Main breakers that size will be tough to come by, don't bother, just backfeed a regular breaker.
Multiple circuits? Somewhat.
Generally NEC allows only one circuit per voltage. So you can have 240V on one pair of wires, and 120V on a second pair.
You can have more circuits for specialized uses which require a separate circuit, for instance the third pair of wires can be used for lighting controlled from the house. I would use that loophole to power the barn interior lighting on a separate circuit from the outlets - that way if the table saw kicks and trips the breaker, you aren't plunged into darkness with your hands 3 inches from a spinning blade.
Re-designate wire? Mostly no.
Generally, wire function is decided by color, and they cannot be re-designated unless they are larger than 6 AWG (at the request of distributors, who do not want to stock multiple wire colors in large sizes). However, neutral (white or gray) wire can be re-designated hot (e.g. to allow 10/2 to feed 240V loads). Also, an entirely bare wire can only be used as ground.
Ok first you're going to have to take the cover plates off the two switches and take photos.
Chances are two of those wires are meant to be hooked together as the "traveler" between switches. Or, perhaps, that one hot wire is power feed to this setup, you need send hot down some line (maybe the other black) for the circuit to work. In that case one of the whites is neutral, the other is switched hot to the bulb.
You can see a bunch of wiring options for 3 way switches at https://www.easy-do-it-yourself-home-improvements.com/3-way-switch-wiring-diagram.html
Next time, take photo before!
Best Answer
The key point in your question is that it happens when the light is off. In this case, the alternating current in the live wire produces an alternativing magnetic filed around the wire which induces current in the floating wire. This induced current is also 50hz that's why you can see it when your multimeter is in the AC mode.
The reason this voltage goes away when you turn on the light is that the second wire becomes live. Since the direction of current is opposite in the second wire its magnetic field is the opposite of that of the first live wire and it cancels the other magnetic field. I.e., one magnetic field is clockwise and the other counter clockwise at any moment in time. So no induced current.
When the light is off there is no current in the live wire so when you put any load on the float wire the induced voltage probably goes to 0.