The short answer is no. As others have said, there are no DC connections from the Ethernet jack.
However if you are ready to make up your own cable, I believe it can be done. First you'll need a PoE (power over Ethernet) device - because your regular Ethernet switch or modem most likely won't be sending 5v over the Ethernet cable. I have set up PoE for an Arduino, but since PoE shields are available for the Arduino, that's not too hard.
I used a PoE device from Freetronics, that cost me about $30 NZ. It has a DC input, which i connected to a wall wart. Then connect a regular Ethernet cable from the router/switch to the PoE device. Also a cable with all cores connected to the other side of the PoE device. The other end of this cable goes to an Ethernet plug, but this only uses the regular 6 wires. The other cores will take power from the PoE device and these would have to be routed into the Pi. You might be able to route them to the Pi's GPIO and so power the Pi through GPIO or you may be better off grafting a micro USB plug on the wires and using that to power the Pi. The PoE device takes a power input and routes this through the Ethernet cable. It also passes the Ethernet data straight through from the switch/router.