First of all: Yes you can, but it won't make you happy.
10Mbit/s Ethernet uses 2 pairs and works quite well over short distances with the kind of cable shown, you just need to crimp an RJ45 plug on it (using the pairs 1/2 and 3/6).
Now 10Mbit/s is not very fast, so you might want to go alternative routes. Assuming The idea behind using this cable is avoiding a recabling, you might want to consider Ethernet over powerlines, this works amazingly well in many home settings, the adapters (Devolo and TP-Link have worked for me) are cheap as chips.
EDIT
As requested in the comments, a few more details about the needed plugs:
Ethernet typically uses RJ45 plugs with 8 pins (1..8) carrying 4 pairs of wires. For historical reasons, these pairs are assigned to the pins this way: 1&2, 3&6, 4&5, 7&8. Older Ethernet connections use only the pairs 1&2 and 3&6.
So for a straight-through cable you need to connect pin 1 of one end with pin 1 on the other side, same with pins 2,3 and 6 and for a cross-over cable you need to connect 1-3, 2-6, 3-1, 6-2.
If the wires in the cable are paired (i.e. always two are twisted together inside the cable's protective mantle) honor this pairing when connecting the cables.
The mechanical process of putting the plug on the cable is called "crimping" and it is easily done with cheap "crimp pliers". If you don't have one, I recommend you just cut a cheap Ethernet patch cable in the middle and solder the two resulting "pigtail" half-cables to the ends of the old cable, using the orange/orange-white and the green/green-white pairs.
No. Both technically and politically.
The technical problem is that the "NBN guy" visited your premises to install a new cable between the node and your house and to install a new socket inside your house. This is usually done by tying the new cable onto the old cable in the street, then pulling the old cable out from the wall socket until the new cable appears inside the house. With the old cable cut at an unknown location it cannot be used to pull the new cable.
The political problem is that you do not have a first socket. It sounds as though it was removed illegally, or you would have been in the Telstra database as having no existing connection. Now that the installer has completed a job report on the unsuccessful installation, you are in the NBN database as having no existing connection, and are very likely to be classified as requiring a new installation not a transfer from POTS to NBN. Which is a whole different (and likely much slower) process.
Your question actually appears to be asking whether you can run VDSL over Cat3. Technically the answer is yes if you want a kilobit per second data connection. Politically the answer is no, as you will then have an illegal connection.
Best Answer
The answer is possibly... but without knowing more about the cable and the layout it is difficult to tell.
In theory, telephony cable can be used to support 10/100 connections (not Gigabit) over short distances. If you can find a telephony cable that is not cut and is a "home run" between the two points you need to connect, and it contains at least 4 conductors, it may be possible.
For 10/100 Ethernet to work, you need to have a straight through connection of pins 1, 2, 3, and 6, and the other pins are not relevant/not used in this scenario.
For the purposes of this, we will say the cable has Red, Green, Yellow, and Black conductors. They may be different colored or contain more conductors, but this is all we are concerned with. If the cable is solid with a matching stripe (Blue with a white stripe and white with a blue stripe twisted together, even better).
Connect both ends of the cable the same to an RJ45 JACK (do not crimp ends on the cable), essentially we are going to substitute color pairs of the existing cable for "standard" color codes in EIA/TIA 568B color code.
In the 568B color code, we need W/O-O/W (Pins 1&2) and W/G-G/W pairs (Pins 3&6) Pins 4, 5, 7, and 8 can be ignored (they are for PoE and Gigabit applications, which we are not going attempt with this type of cable).
Now substitute... If you cable is Red/Green Yellow/Black, connect the green to Pin 1, the Red to pin 2, the black to pin 3 and the yellow to pin 6, do the same on both ends and connect with a standard patch cable to the router and computer. If the color are different, just substitute as necessary.
Now, the reality this might work perfectly at 100Mbps, or it may error out like crazy... it may function fine at 10Mbps or fail just as miserably... Or it may not link up at all. There is no way of knowing until you try it.