I keep most of my seeds in the fridge. They are in miscellaneous paper packets (I just started using glassine) and are grouped by category in ziploc bags. I used to use plastic tubs, with tidy little labelled dividers I made out of cardstock, but I decided they take up too much space. There may be more seeds than food in there, and the bags are compressible before zipping

. After 10-15 years, germination is less, but it's no problem at all after 5 years.
Seeds that I have larger quantities of, like brassicas and favas, I just keep on top of the bookcases in the living room. If I have saved a lot of seed, I don't care about germination rate

. I did find bugs in one paper bag this year (10-15 years old), so I will be more consistent about putting them in containers now.
I also don't clean my seeds, necessarily. For some varieties (brassicas, again, are an example) it just doesn't seem worth the trouble. I save whole stalks/pods, and crumble them into the bed. Or, if only starting a few in trays, I separate out the seeds at time of planting. A side benefit is that the chaff makes it easy to see where I have planted.