I've lived in many a friend's and relative's basement.
First thing's first, think small, just carve out a corner of the basement and make sure it's where you
have a window. I mean if they're gonna let you take over the whole basement then fine go for it.
But if not, I recomend taking over just a corner for your personal space, and then fixing up the adjacent
area for a house hold "chill spot". This way you won't be viewed as taking over the whole space.
If it has unfinished stone block walls, Paint that shit! You don't wanna be breathing in that stone dust
while sleeping, it will fuck up your lungs. Just paint the corner where you'll be sleeping. Get a cheap bed, or
reject mattress from the back of a mattress store. If all you got is the mattress, build a crude bed
frame by laying down some cinder blocks with plywood boards on top. Hang some old curtains, blankets, etc
from the ceiling around the bed to form two walls, this will give you some privacy and prevent any drafts.
This was what I pretty much did for the last basement situation I was in a few years back. I got most of
my furniture and other items from a combination of thrift store finds, neighborhood trash and what was
already down there. I found a military foot locker on the curb, they are good for keeping your private shit
safe. I already had an ammo box to keep my food in(rodent proof) and a portable clothing closet.
I do recomend the metal box for storing food. And you'd be surprised what you can make a mini
closet out of, like a cedar chest or foot locker opened and turned on end with a wood rod rigged in it.
For the chill spot I just put down some old rugs, placed various chairs that I found, a coffee table,
some ashtrays, homemade posters on the walls, a book shelf full of used books, etc. My host liked
what I did so much he gave me the go ahead to slowly work on the rest of the basement... rent free

Everyone in the house seemed to really like the chill spot which was between my space and the laundry room,
and everyone respected my private space beyond the curtains which they nicknamed "the Arab tent".
If your amount of personal space is limited to practically just the area for your bed, you can make the
curtained off bed area double as closet space by attaching some wood rods from the rafters above where
you're sleeping, just hang your clothes there, safe and out of harms way. You can also store a lot of your
stuff in containers pushed under the bed.
I know every basement situation is different. This info is just from one of my experiences, I hope it is of help.