My current character obtained a large amount of wealth during his many and wonderful journeys, but one problem plagues him. He's running out of space to store his collection, and this is a problem because I collect many things. Elder Scrolls collections are fun to make, in my opinion. I've run out of space in two locations, now, and I really just want to be able to store all my stuff in one place, my Vivec apartment. Indarys' Manor is nicer and larger, but I like my cozy little apartment in Vvardenfell's greatest metropolis, and Indarys' Manor has too much ash and rock yet too little plant life for my taste. Are there any mods that allow containers to store endless amounts of loot, as in Oblivion?

One thing you could do is open your apartment in the construction set, double click on a container (chest, crate, box, etc) you already have in your apartment, and then change the container weight to 999,999. I think this is the max you can set in Morrowind. There is one thing to watch out for though....I think it's possible if you alter the weight of a container like that, it might alter the weight of every container like it in the game. I think you can make it a unique ID somehow though so that will not happen. I'm pretty much beginner at modding so someone can probably amend my suggestion to be better.

I'm worried that I'll mess something up.

Be sure to remove everything from a container prior to altering it in the construction set.

Lorsk is correct--to avoid changing the capacity of every container of its type, you need to give yours a unique ID. What you actually do in the CS is copy the container, edit the copy to have the new ID (and of course the increased capacity), delete the container from its cell (your apartment), and place a new container with the new ID in the same spot. Any construction set tutorial will tell you how to do all this.

Well, while this doesnt have infinite storage, it probably has some of the most you will find in any mod. Alladin's Bottle Revamp: http://planetelderscrolls.gamespy.com/View.php?view=Mods.Detail&id=8212
This has 24 manniquins, many tables and shelves, probably close to 2 dozen chests/lockers to keep stuff in, and this is just in the display/storage area. There is more storage in the bedrooms, etc. Now while I havent looked at a ton of house mods or anything, you probably wont find very many mods with this much storage/display.

There are also mods out there which give you a Bag of Holding, which allows you to carry around a bunch of stuff without counting against your encumbrance limit. The one I'm familiar with is in Uvirith's Legacy. Note that a) it's large but not infinite, and B) you can't get it until you become Telvanni Archmagister. I know there are other mods with BofH, but I don't know which ones.