Sell them on ebay and travel Europe for 2 years.
Seems to me the real problem is not “storage” but knowing what you have and where to find it. Alphabetical is not practicle. You end up with too many of “S” or “T” or something.
Cover a wall with peg board. Set up a 10x10 matrix. This will give you 100 hooks poking out. Number your CDs with a 3 or 4 digit number (doesn’t matter what order- you’ll see why) Catalog them by last 2 digits. Print a master list out however you want to catagorize them with a “peg number” that matches the last 2 digits. Thats where it hangs. You should only have to look thru about 50 cds max to find the one you want. New additions are distributed thru out the entire collection without regard to artist/title/genre. I developed a similar system for retail special orders. Instead of having 26 shelves for “name” (of which everything ended up on the “S” for Smith shelf and the rest sat empty, mocking you!), I had 10 shelves, all moderatly full, that matched the last digit of the special order. Worked like a charm.
Problems: 5000 cds will weigh a whole lot! Peg board or your wall might not take it!
They won’t have any protection, and will be rubbing against each other every time you mess with them.
If I was you, I would cull out the ones you don’t really listen to anymore (and that has got to be a bunch, there are only 24 hours in a day), maybe burn the one good song off some others onto a “greatest hits” collection and shit-can them.