The motor you got is a unipolar stepper motor. The five wires are one end of the four windings inside the motor, plus a common wire that connects all the other poles together.
Bipolar motor offer more torque than unipolar, at the expense of a more difficult control. 6 wires unipolar stepper motors can be converted to bipolar motors just selecting the correct wires. I am not aware of any method to convert a 5 wires unipolar motor to bipolar that does not imply tearing it apart (which is not really a great idea).
The L298 is not really made to drive this kind of motor, it is best suited to drive bipolar motors (can still drive it tho, please read the next paragraph). Good news, driving unipolar steppers is rather simple. You basically have to connect the common wire to you V+ rail and connect the other wires to GND via BJT/MOS. Don't connect them directly to the Arduino! It's the perfect recipe for fried chips. You can either choose to use discrete transistors or arrays like the ULN2003.
If you still want to use the L298 to drive the motor you can exploit the lower transistors of the H-bridge. Connect Vss and the enable pins to 5V, Vs to your positive rail voltage and pin 1, 8, 15 to GND. Then connect each phase to each output and the input to 4 pins to an Arduino/microcontroller. Leave the pins normally on and turn one off in turn (1->2->3->4->1...) and your stepper should run! This works because when you leave the pins on, the upper side of the H-bridge is powered on, thus shorting the winding. When you turn one pin off the lower part of the bridge will actually turn on and let the current flow to GND, closing the circuit and powering the winding. Using this kind of circuit you could also omit the freewheeling diodes needed to discharge the windings (as said before, you short them, allowing current to freely circulate).
EDIT/TLDR: Connect the windings to the output of the L298 and the common one to the positive rail. Keep the pins high and turn one sequentially off, you should see the motor move.