2009 Chrysler 300c will not lock or unlock on both remote keys .So key failure seems unlikely. Would it be the remote receiver in the car ? What about the receive antenna ? Is this a common problem ?
I don't know that I'd call this a "common" problem but it's certainly not unheard of.
First, be sure to check that both of your remotes have good batteries.
Then, if neither are working to unlock the car, the most likely problem is that the car has "lost its mind" with respect to the IDs of the remotes. Some locksmiths have the equipment to program remotes into the car but usually you will need to have a Chrysler Dealer do it.
It's also possible that the receiver unit itself has failed. Again your dealer will likely have the equipment to test it and install a replacement, which likely requires special tools.
I'm assuming that the car's battery has charge, if not then to quote Mythbuster Jamie Hyneman, 'Well there's your problem!' If your battery has charge and the car runs normally then it's worth replacing the fob batteries first before you do anything else. The coin batteries in the fobs are very cheap and simple to replace, and it is possible that they've both gone flat at the same time. Worst case you're out a few bucks, best case is problem solved. Next, check that the fuse which powers the system is good, and making good contact.
If those don't help the next step is reading the OBD codes, the diagnostics may point you in the right direction. Look for other issues like window controls, climate controls, and similar, if you have other problems it may be a problem with your body control module.