Yes, you can make this cable work, but you are unlikely to get good performance, particularly if you try and run faster than 10Mbps. I would not use this cable if I could figure a way around it.
Basic Ethernet only requires 2 pair to function. You should map two of the pairs in this cable to the colors/pins needed for Ethernet transmit and receive (typically Orange and Green??) and terminate according to the instructions in whatever jacks you get. You can get jacks that don't require special tools to terminate (example) .. they should be fine for low speed applications.
The only other thing I would do is force the NICs and switch ports to 10Mbps in order to minimize problems.
REMINDER -- this is a last resort solution. If at all possible, get good CAT5e cable and have it installed by pros.