We had a similar issue with the kitchen in our rental apartment. Almost zero storage space and not much counter-space, either.
We ended up getting two ikea "islands" and putting one in the corner of the kitchen and the other just outside the kitchen to add in a prep area (the kitchen as-is has no prep areas, annoying). We also got a chest freezer and occasionally use that as a counter for chopping veggies, too. The islands all have shelves underneath and we use that to house appliances we use often because there is no room in the cupboards.
For the food storage, we got 2 large wire shelves and put them just outside the door to the kitchen. That's where we put cans, baking stuff, cereal, noodles and grains, etc. So we just have to do two steps out the kitchen to grab anything we need, and it isn't too much hassle. The shelves themselves are thin but tall, so they don't stick out much and are unobtrusive to our walking paths through the house.
So it isn't the most elegant or ergonomic solution, storing our pantry items just outside the kitchen has worked out really well for us. I've just stopped caring that any visitors can plainly see everything we have "in" our pantry (it is like the first thing you see when you open our front door lol).
For more freezer space, your only option is to get more freezer. I do recommend a chest freezer--you don't have to put it in your kitchen. My parents have two huge chest freezers in their basement and one in the garage. Whenever you need something, you need to take a trip to get it, but at least it is out of the way. DH and I have one smaller chest freezer, its only about 10 cubic feet, so its slightly less obnoxious.