What can give me near-infinite storage?

I have a lot of things. I need a place to store them all.

Barrels store only one thing, but a lot of that one thing.
Applied Energetics systems store 63 types of things per drive.
Chests, are well, chests. The best is the diamond one.

I am a greedy person and want a way to store all my things without taking too much space. What do I do? I am using the Ultimate pack in Feed The Beast.

Deep Storage Units.

A lot of space is taken by the same thing spanning multiple 'slots' in an ME drive. Cobblestone is the worst offender in my system. A single crafting will net you 4 DSUs, and putting the 4 biggest volume items in DSUs tripled my space when I got full. They're cheaper than drives, if you have one item that you need to store a LOT of.

For easy management of the contents, I'd recommend an ME network using a Storage Bus. If you right-click it with a wrench while sneaking, you can set the Storage Bus to have a higher priority than any ME Storage Drive. The movement is bi-directional, too, so you can extract items using your ME Access!

If you are using an ME Export Bus to move contents to the Deep Storage Unit before hooking it up to your ME Network, take care to attach it to an input face. This can be configured using the DSU GUI. A Storage Bus, however, is able to both deposit and withdraw using an output side.

Immibis' tubestuff mod adds a Black hole chest which can (very nearly, it has up to 2^31 pages) store an infinite amount of items in it so that is definitively the most compact way possible to store (effectively) all your things. However the configs disable it by default so you would need to be able to edit them to make it available in survival mode.

It has also been known in previous versions to be quite buggy and easily capable of losing all of your items, I don't know whether this has been fixed. The chest is also not sortable so if you stored all your stuff in it you might never find some of it again.

Other than this your next best option for compact storage is an ME Drive, which can hold up to 10 me storage disks for up to 630 item types in a single block, which has the downside of requiring an ME network to connect to but the associated benefit of being able to hide the storage somewhere else and only having a few access blocks visible in your base.

An alternative approach (and the one I like to take after I've got an initial base setup and enough resources available) is to instead of storing all the things all the time just store enough of the things for your (optimistic) near-future needs and have an automation setup that will replenish your stocks as you use them. In short slow down your collecting of all the things so that you don't need as much storage space at once.

You could use a Black Hole Chest which can store indefinitely