Stacking boxes is not a good idea for something so fragile. Here is a reasonable solution from IKEA that will keep your LP’s in pristine condition. Store them in KALLAX and wrap it with polythene plastic cover bag from U-Haul.

To go higher, you need to have a strong shelf a few inches above the bottom stack of 3

say some tall saw horses with long thick plank, then you can stack another 3 boxes on that plank above the bottom 3.

figure some way to independently support the top stack.

You might consider putting the LPs on shelving racks in the storage unit. That will give you access to them, you can sort, weed, sell, give away and then decide what to keep for yourself. Occasionally grab some to listen to.

I bought chrome wire shelving, added wheels to the bottoms, this type of thing, look around, various sizes and prices. I had wood shelves with dividers to put on the wire shelves, you would need to add a smooth shelf on top of the wire to avoid damaging the bottoms and so they can slide in/out.

Smooth metal shelves might be better, but they need to be strong, if not: several 30" wide units better than 36" or wider

Get a bigger storage unit.

Have a look at my main virtual system. I have about 2,100 albums. These white cubes come in singles and doubles. They are not expensive. 

Whatever the box or cube system, it is a good idea to shift the records a bit from time to time. They should be store vertically, not lean to the side, yet not so tight that they cling on to each other.

The collection was in my parents' FL condo for 30+ years. Pop had them on some cheap graybar shelving. When I packed them, I made sure the boxes were snuggly packed so they wouldn't shift, lean or sag. 

I plan on using the Kallax system in our next home as the storage solution. The thought however of doing that in a storage unit somehow doesn't seem practical.

Thank you all for your thoughts.

Happy listening.

Florida. Check for mold.

The collection was in my parents’ FL condo for 30+ years.

costco shelving, mine was Gorilla brand…..

Use the black totes with yellow lids or the larger black totes with red lids (with or without wheels) from the big box hardware stores. We use them for theatre lights storage 4 high, and they are heavy. After moving the LPs to your new place, you can still use the totes for other storage.

Good advice given. 
Pay attention to MOLD. No moisture, humidity. Once mold starts, it will spread. 
 IKEA shelves are excellent. If you set them up in a storage unit, you can easily disassemble and use them somewhere else. If you’re inclined, Craigslist is a good source for used IKEA shelves. If they’re used in a storage unit, who cares if they’re beat up. 
 

I use Ikea shelves for LP storage.  They are just the right size and can handle 50-100 lp's per section of shelf.  They were inexpensive at the time. 

Depends on your card board box strength. Not all boxes are created with the same amount of stacking strength per weight. Lots of different storage boxes out they're for sale. My question is do you pack the records on end or flat on their side for long term storage??

@68pete they're on end, of course! The boxes are heavy moving boxes, all new. 

Regarding mold, the condo was airconditioned and well maintained. And as I mentioned, everything is currently in climate controlled storage.

Happy listening!

@ghdprentice 

Where do you get those cubes? Is there any need to anchor them or will they stack fine as is?

@ozzy62

They are available all over. No anchoring is necessary. Gravity does a great job.

Here is a reference on Amazon. But worth researching. The doubles are great for stacking high.

Use this to search Amazon.

Way Basics Vintage Storage Blox Cube Organizer Shelf 

@ghdprentice If you paste formatted text into the editor when the Source button is selected, it will appear as plain text. 

Thanks 

@noromance 

Thanks.i screwed around with it for a long time and finally gave up. I’ll see if I can remember that next time.