I've lived in those conditions in the past. It's hard but it can be done. Some mediums are more practical than others. It's possible to work large in mediums like pastels or watercolor where you're working on paper and finished works can be stored in either large plastic under-bed tubs or in the same big cardboard boxes that full sheet papers and boards come in.
Organize ruthlessly. Get a lot of plastic tubs and boxes with lids, some with locking lids. Keep all supplies organized by medium so that if you do more than one medium, it's a matter of opening that box and taking out a canvas to work on it.
Oils are not a good choice for painting in a small space, especially with poor ventilation. Consider switching to mediums that have the least fumes and cleanup from those that involve using thinner. Acrylics at least you're only using water, the same with watersoluble oils. Acrylics dry fast, so what you have are finished dry canvases that can be stacked up against each other without damage.
Living with wet oil paintings may be hazardous to your health and subtly demoralizing. If you're constantly smelling thinner and have a low grade headache you don't notice, every time you paint you could be conditioning yourself to avoid painting and the consequences of living with wet paintings.
Consider working on primed panels and boards rather than on stretched canvas for space even if you're still doing oils or acrylics. While gallery style canvases don't need framing to sell in a gallery, they each occupy 2" deep space when the art's finished and while wet need another inch to dry before the face touches another painting. If you can make or get a wet painting rack you can make those take up less space. I saw an ad on Jerry's Artarama yesterday in my email about an interesting wire rack that starts at about $50 for the smaller wall or table mount one - the big one rolls around on wheels and is more suited to a large space. I'm thinking of getting a small one and wall mounting it so that I can do works on canvas and have a way to store them while they dry or after they've dried and before they sell.
Consider working small. One way to do "large paintings" is sectional - get those packs of four or nine 12" square canvases and divide your planned painting so that it takes all of them to create it, then hang them evenly an inch apart in the display. That's popular. Get creative with things like that. Do tryptichs and diptychs so that separate panels or canvases are smaller and add up to a larger composite work.
Or work on paper. You can get big sheets of glassine and store big works on paper flat under the bed in a cardboard box provided by Blick or Jerry's when ordering the paper online. I use an inexpensive cardboard portfolio for full sheets but I don't actually do full sheet sized paintings, if I did I'd have to create an under-bed storage for them. A lot of them can be stacked flat with glassine between them to protect the painting's face.
Pastels are a good medium to explore. Pastels give instant gratification. They have no drying time and are always reworkable. I'm drifting more and more into pastels over the years, both oil pastels and dry pastels (the chalk texture ones, don't call them chalks unless you're going through an airport). Works on paper store and ship better than canvases, they just take less effort.
One relatively new product may be really helpful to you - Pan Pastels handle more like paint. You use Sofft micropore sponges rather than brushes, but they are cleaner and don't leave as much dust around as stick pastels, which work well with them as accents. Pan Pastels are cheaper in sets than open stock, what I recommend are the Painters 20 color set to start and a mixed bag of tools or if your budget won't handle that, the 10 Painters set because it's a good palette. All 60 of the other colors are very useful convenience colors, gradated value mixtures of white (20 tints) or black (20 Shades or 20 Deep Dark Shades) with the pigments in the 20 Painters Set.
You can find out more about them and see a lot of good paintings at the Pan Pastels site.
There are videos on how to use them. I've got the full range because the convenience colors really are very useful, they speed the process for me. Most of all I can use them in close quarters with very little cleanup and no liquids. I don't use fixative with them.
If you want to try stick pastels and want to use fixative, use SpectraFix because all the normal spray can ones have toxic fumes that cause breathing trouble and well, headaches and a nasty odor. SpectraFix smells like you poured yourself a vodka based drink - rather a fresh, pleasant odor that's not distracting or toxic. Really try to avoid toxic materials in small spaces.
Pastels, oil pastels, watercolor, gouache, colored pencils are all mediums that do need framing with glass. That can be a bit frustrating when galleries want you to frame your art for them. However it can be worth the cost to do it when the problem of storage is reduced and also the problem of living with drying paintings. Some sales can help. In your living situation I would do the art, store it unframed, sell some small format works and ship them unframed and save up the money for a gallery arrangement so that when you're ready, you take the art to a framer and have all the work done for you and delivered to your gallery. Don't even bother storing anything framed unless it's returned by a gallery. You don't have the storage space.
Online buyers are perfectly fine with taking it to their own framers. Online markets are always in reach too, by way of post office or UPS or FedEx pickup. Pass on the shipping costs plus a modest fee for your labor packing it and the packaging to your customer. If they're local they can come over to pick it up in person and save that money.
I've lived in small spaces most of my life. Mobility issues and chronic fatigue make painting large impractical for me. But if I wanted to go medium large like 18" x 24" or 16" x 20" there are pads of good archival watercolor and pastel papers that include their own supports. I like watercolor paper on blocks bound on all four sides - it's just easier to avoid the cockling and I don't have to do the labor of stretching and taping it to a drawing board and all that prep. I do not have the body energy for most of the usual prep large painters have to do.
Use short cuts intended for plein air painting. Use easels and boxes and storage solutions intended for plein air. That means it takes up less space too. It folds down much smaller and can be put away completely in an organized way when not in use. I've been doing this for years. I get excited about some sort of plein air kit and have all these fantasies of painting outdoors... and then what really happens is the little kit gets a lot of use sitting right next to my computer where I don't have to get up to get it but it's not in the way when I'm not using it.
Sketch daily. Start the day with your sketchbook. Plan ambitious works when you get ideas for really big paintings. Don't ignore those ideas, you won't be living in your family's house for the rest of your life.
Hang out on WetCanvas and participate in challenges. For me they often result in salable paintings. Sell them online. Nothing encourages like success, especially when it was for something you did because you liked it and you happily discover how many buyers like it too. Do it your way.
A lot of challenges say "try something hard, go outside your comfort zone."
There are times when "Return to your Comfort Zone" is the best advice. Choose a subject you're already good at. Use your best medium. Indulge yourself in appreciating your mastery of that medium and that subject. Stretch in little subtle ways with it. You'll discover those stretches in process, follow your artistic impulses. Most of all do the works that make you happy - somewhere out there in the millions of art buyers there are people who share your tastes and you'll be a happier artist painting for them than trying to follow trends.
Daily Painting is a good direction for working small. Start a Daily Painting blog and maintain it - work small, most of them do so it can be finished in one day. Then make sure the daily paintings are available online at eBay or Etsy or some other marketplace that you will get paid online and ship anywhere in the world at cost. That market never sleeps. It also doesn't run dry because collectors who like small works usually have enough wall space for lots of them and may rotate their collections when it's full.
They also always have another paycheck and are fond of repeat buying when they discover they like your work.
Doing paintings in series of a favorite subject is good. But don't stricture yourself too far. Experimenting and trying new things is good too. Having ten of a particular style and subject is enough for a gallery but try not to sign anything that makes that exclusive - you want to be able to market your other stuff too.
I hear you about house arrest in the suburbs when you don't have a car. I lived with my family for five years in Kansas and Arkansas without a car. That can be maddening. Either save up for a car so that you can get out of the house or save up to move to a city with good transit and a thriving art market. My daughter got her dream house with 20 acres where she could keep her horses (she has three, all adopted, people have more trouble re-homing horses than cats or dogs even if they're an expensive matched pair of white Arabian geldings.)
When she moved there was no longer a spare room for Dad. So I had to move sooner - but I'd been planning to move back to San Francisco and saving up for the move on my Social Security. I moved last August. It was brutal, both physically and financially, but I managed to pack all of my art supplies and ship all the essentials before I got here, found housing online and by phone, got things sorted out. It took longer settling in than if I wasn't disabled but now - now I'm free. I've even got Paratransit since I found out I really can't walk far enough to use the normal bus service.
Save up to get out of that situation, either moving or getting a car or both depending on your lifestyle choices. I chose the city because I don't want a car even if my eyesight is still correctable to legal driving level. It's like supporting another person in expenses - payments, insurance, gas, that's a huge expense on top of living expenses compared to being able to get around in a city by buses and cabs. Sometimes cabs are cost effective, depends how close something is and how much instant convenience matters.
Cities also have more choice of galleries so it'd be easier to find the right one that's a good fit for what you paint.
Save up for that. Sell your art and build your savings. It makes a difference to know you're leaving - it did for all the time I saved. And draw or paint daily, that makes a difference too. What you can't do now will be possible in future.