Here are my thoughts on safes and safe deposit boxes:
Safe deposit boxes (i.e., note that they are not called “safety” deposit boxes by any banks in the U.S.) are not the way to go. Many banks do not run A/C regularly in their safe deposit box areas outside of their working hours and most have a dollar limit on what can be stored there. If you look on the old sports forums, you can see decades worth of horror stories: mixups, flooding, humidity issues, $10,000 insurance caps, etc.
Similarly, home safes can cause humidity issues and don’t adequately protect against fire. Most commercially available safes are UL Class 350, meaning that their job is to keep the internal temperature of the safe below 350°F (175°C). However, the cheapest safes will only do so for 30 minutes at most and the best ones available will only do so for roughly 2 hours. House fires burn at 1,110°F (593°C) on average and can engulf an entire residence in less than 10 minutes, so any delay in firefighting will fry your cards.
If you think 350°F is OK for an hour or two given that paper burns at 451°F and cardboard burns at 427°F, you need to consider the melting point of plastics (i.e., cards sleeves and cases used for grading). Unfortunately, most plastics melt far below 350°F. So even if the safe is doing the right thing by protecting your valuables at ≤350°F for 30-120 minutes, your cards will have been damaged/fused with the plastic that protects them for some time. Also, none of this takes into account the potential for smoke damage and the issues that safes can have with regulating humidity.