When I moved a year ago, I had to pack my 3000 records so they would survive two moves, first from our old house to a storage warehouse while our new house was being finished, and then 3 months later from the warehouse to the new house 160 mikes away. I packed them myself using about 35 cardboard boxes purchased from the moving company. These were labeled book storage boxes and were about the same size as the U-Haul boxes described by others. Fortunately the moving company had a good number of used boxes they sold me for $1 each. They had been used a few months earlier to store books when a library was being renovated. Somewhat to my surprise, the boxes looked almost exactly the same when they were delivered after two moves and three months in storage. And the records were exactly the same as when packed. Maybe I was just lucky.
I did not trust the moving company with my equipment though. I moved everything in a U-Haul truck to a rental storage unit where it stayed until the new house was ready. That was a lot of work but it all worked out well.