Our pioneer grandmothers also stored root vegetables (potatoes, beets, carrots, and turnips)  in aptly named root cellars. These were usually dirt-floored rooms dug into the ground with a door on top. Vegetables were stored in baskets, laid out on shelving, or buried in sand to keep them from drying out.  Most basements and many garages today however are heated, making storing root vegetables there no different than storing them in your kitchens. 

Today, we usually store carrots, beets, and turnips in the fridge. Potatoes are best taken out of the bag they are purchased in and put in a bin or drawer in a cooler part of the kitchen. Onions can also be stored in a bin or drawer, but one should always be in the fridge before you need to cut it. Storing onions in the cool reduces the amount of tearing you will experience when you slice or chop them, without cutting down on their flavor.
Garlic, however, should be kept in a dry place and used as needed. Fresh garlic mellows out in the fridge, losing its wonderful flavor and pungency. A clay garlic container, which is breathable like a flower pot, is best. Ceramic ones should have air holes.
Tomatoes are usually picked ripe. But if a hard frost is coming, you can glean all of your tomatoes, no matter how green. Some people swear that putting tomatoes on the window sill is the best place to ripen them.  You can also put them in a bowl with an apple to let them ripen.  The apple gives off a gas that hastens ripening. You should rotate the tomatoes top to bottom every few days to make sure they are ripening evenly. Sometimes a tomato on the bottom may ripen faster than the ones on top. You want to rescue it and use if before it spoils. I’ve gleaned hard, green tomatoes in late September and eaten fresh, ripe ones for a month.
Pumpkins and winter squash, like potatoes, should be kept in a cool place. The fall is a great time to get them at the grocery store or at a farmers market. If they are kept cool, they can last several weeks. If you are in doubt, though, you can always cut them up, cook them, and then freeze in containers or bags.