Sure it can be made safe , or let's just say as safe as it was when it left the factory back in 1958 . At the minimum , the big capacitor soldered to the speaker has to go , as well as any other "wax paper" type capacitors (this includes any "Bumblebees" that may be lurking about) . Since you said "I don't have a clue when it comes to actual electronics" my best suggestion here for you is this ;
Since this is obviously an "heirloom" piece for you because your not likely to find another as close to your family's original radio as this one you've now found , put it away for a bit , pick up some other (
or two or three) "AA5" tube radios and learn how to do a "recap" on them as your learning experience . You'll of course need to learn how to solder , no biggie there , millions of folks can do it and so can you . A search here of "recap" will turn up all kinds of great reading and although it may seem daunting now , you've chosen one of the easiest sets to repair to fall in love with , the five tube "All American 5" being one of the most simple designs .
I will show you a picture of some recent capacitors I replaced , these are mounted to a proper terminal strip and are 105c temperature rated . In your case , where you have a circuit board , there is no reason why some new capacitors couldn't be soldered back onto the board in the spot that the original (C1) was .