I see spiders regularly in my house (mostly in the kitchen and bathrooms). There are no flies or mosquitos around this time of year and I believe they are obligate carnivores. There is an occasional roach, but they are enormously larger than the spiders (which rarely exceed a cm. For the last couple years, we had an infestation of nut moths (I had a sealed package of beans from a local company that was full of them, so that was obviously the source) but I haven’t seen one in a couple months. So where are the spiders getting nourishment from?
Many of them die in the fall, leaving behind an egg case for next year’s brood. Tarntulas (not technically spiders, IIRC, but mygalomorphs, which as I understand it are awful dang close to spiders, but about half a step back on the evolutionary ladder) tend to live in tropical climates where there’s less of a seasonal change.
Maybe I didn’t make it clear. These are tiny (maybe 1cm) spiders that I see regularly in the kitchen and bathroom all winter. I doubt that they leave the house. Presumably they need water. But can they live on microscopic dust mites? That would be the only source of food that I can think of.