Lots of people have different ideas for storage of firearms and barrels.
I have seen a lot of wood stocked firearms that were put into long term storage in crates and stored on their sides. Everything from vintage matchlocks to M14s and more recently SKS rifles by the crate. The seem to be no worse off for age as long as they have been stored properly in dehumidified areas.
The shipments of replacement barrels I have seen, such as those offered by Tradex, one of our banner sponsors, were stored and shipped in flats on their sides.
I saw a bunch of FN SLR barrels in heavily waxed coverings along with L42 barrels in similar coverings that came into a shop I worked for as surplus and they were all bundled together in a half dozen to a bundle in one heavy press board tube with manufacturer's numbers, dates and the description printed on the tube.
IMHO, just store them carefully. Fill the bore with cosmoline or similar grease and cover the outside of the barrel in similar material, then wrap it up nicely in a plastic sleeve if you plan on storing it indefinitely.
If you are planning on using it relatively quickly, coat the bore with a good preservative as well as the outside and store it in a dry place with no outside weight against it. All should be well.