Can anything be done to shield a record turntable from picking up unwanted radio transmissions?
Serious radio amateurs know how to fight RF interference.

Just adding a shield and taking care of proper earth grounding might help but is probably not enough. Frequently the RF signal is picked up by the incomming mains cable and so measures need to be taken in that area. 

In your case the power supply is realised with 4 diodes and a capacitor. (Top right on the picture. To make the apparatus insensitive to incoming RF signals a little change of the simple power supply could be effective.

Try to place (have placed) small capacitors over the 4 diodes with values between 10 - 100 nF. The schematic added is for 230VAC but is also valid if the incomming mains is only 110VAC. Even in situations where there is no mains transformer involved.

Why?

In many situations the problem is caused due to the increased sensitivity of semiconductors for HF and VHF signals. Even a diode as 1N4007 is capable to demodulate such signals.