I keep a tray on the bottom shelf of my fridge. This is where all raw meat goes that needs to thaw or is bought to be cooked later. If it leaks, I just clean the tray and put it back in. No more having blood get all over the refrigerator.
Unfortunately, plastic trays that don’t have open holes for carrying handles are hard to find for a decent price. Something similar to a lunch room or an old fashioned Sonic Drive In tray is what you need. If you can’t find one like that, use anything that works for you. Here is a link to a round one what would work. You may already have something you have in mind.
You could use a Tupperware or Rubbermaid dish or anything that works for you.