First off... an unusually long winter is its own refrigeration. You don't need the Frigidaire when it's -10F outside. (Of course, waiting for it to come, things would spoil.)
I'm only a lowly agriculture student and hardly an expert, but even today with mechanized agriculture, it's tough to grow enough hay for livestock for a 3-4 month winter. Farmers will buy it shipped in from other places. If you figure a 5-7 month growing season is typical, and a 4 month winter, well obviously we're breaking even (as a whole), and putting away a little extra (when droughts happen, we can often ship in hay from other places in the summer without using everything we'll need for the coming winter). But this extra is going to be 50%, maybe 100% or even %150-200 (in a good year, and only locally) of what we need.
Boosting the growing season would help. If in 5 months you can grow (mostly) enough for a 5 month winter, then in theory a 10 year summer might (mostly) allow enough for a 10 year winter. Except... hay isn't just dead grass. It's semi-dried plants and if it dries out too much it loses much of its nutritive value (and if it stays too damp, it gets moldy!). Even if you were rotating, eventually you'd be storing this stuff longer than a year or two, and you'd see lots of spoilage and even it drying out and crumbling to dust. While the winter can refrigerate, the wait for it can be much too long. I don't see how decade long summers would help to save enough away for even a 2 year long winter.
And that's just for livestock.
Humans can't eat hay. They need fresh vegetables at least every couple of months, or scurvy sets in. It will kill, it will kill much sooner than the end of a 5 year winter. And while you could freeze and thaw vegetables, the extra-long summer doesn't allow you to do this except at the very last moment. So you can't save up 10 years worth of onions and carrots and lemons and whatever else.
Now, other objections can be waved away. Seeds wouldn't last 10 years in our world, but in a world where plants have adapted to this, I assume most would. Our orchard trees would need periodic winters for dormancy, but that will have been adapted out in their world.
But this rest? The inhabitants can't have anything resembling human biochemistry I'm afraid. And on top of that, the logistics would be intractable even for our own level of technology (remember, the government was able to plan to save a few hundred thousand out of hundreds of millions for a short 18 month nuclear winter).
So, the answer is "magic".