In addition to canning, freezing and drying; we like to "store" some veggies right in the garden. We usually plant leeks, turnips, carrots, daikon radishes and cabbages in a patch together (closely spaced) in late summer for winter harvest. After cold weather sets in we cover the area with loose straw (about a foot deep) to prevent the ground from freezing. We're then able to pull the straw and snow back and dig some fresh veggies all winter long. The root crops last until spring while the cabbages and leeks carry well into February. Mice and insect damage is very minimal and the crops seldom freeze. There's nothing like "fresh dug" carrots when it is 20 degrees and snowing out.
This method also works well with spinach, chard and kale but the time frame for picking is shorter. Or you can just leave the straw on all winter and get an early crop in spring before the ground is even workable!