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.
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.