Could I buy a pair of JBL LSR305's and plug in a turntable with RCA, and also a Chromecast audio via AUX? Their doesn't seem to be an AUX port on these speakers. Would some type of adapter work? And could I have the turntable setup via RCA and the Chromecast audio setup via AUX at the same time?
I think you are mixing two different problems that make your question difficult to understand.
As you can see, there is no RCA plug on your speakers. But you can notice a 6.35 mm jack plug.
Even if you find an RCA to Jack adapter, the output level of your turntable will be too low so you will need to apply a pre-amplification, then an amplification and finally you can send the signal to your speakers. Sometimes the pre-amplification and the amplification can be done in one single device.
If you want to plug an additional device to your speakers, I suggest you a mixing table that can do all the work. I'm not sure, but I think Chromecast audio output is HDMI supposed to go on a TV, but maybe there are some devices that will convert the signal to analog(?).