Yes, it's possible. But I guess it depends how you're affected.
I'm 41 and I've lived independently for years. I lived in children's homes as a teenager, so I haven't lived with my parents since I was 13, so yes, it's very definitely possible. I've lived by myself, I've lived with a couple of boyfriends, I've lived in student halls, and I've lived in shared houses. I'm currently living by myself.
The worst thing about living independently, from my perspective, is that I suffer quite badly from executive dysfunction when it comes to managing my finances and bills and expenses, I'm quite useless, domestically, in that respect. (It's weird, because in previous jobs, I've processed expense claims, sent out invoices, chased invoices, processed and paid invoices, but while I can do that in a work environment, if the systems are all set up, I'm seemingly unable to do that for myself.)
If you're not sure about leaving your family home and living by yourself, how about living in a shared house with other people? Did you live in student halls throughout your studies, or did you live in any shared houses? Lots of young graduates start out living in house shares in their twenties instead of buying or renting a place of their own, not just because renting a room is usually cheaper than renting a studio or one bedroom flat. In fact, for many of my friends who moved to new cities for jobs after graduation, they made a point of renting a room in a shared house instead of renting a place of their own, even if they could afford to, because they wanted to meet new people and make new friends.
There are potential problems, though, if you're not particularly sociable if you end up living in a kind of 'party house' where people are socialising and going out a lot. But some houses might have postgraduate students or people who take their work more seriously and who want a quiet life at home.
Adverts will often give clues about what kind of house-share it is, for example they might say something like they 'work hard and play hard' and that probably means they go out a lot or have friends round, and they don't care whether they're working the next day or not, whereas if they say things in an advert like they tend to curl up on the sofa and watch telly on a school night (i.e. if they have work the next day), but they like to go out socialising at the weekend, then if you're not very extrovert that might appeal more.
It's possible that you might move into a house share and find a mismatch between personalities, in which case, you might end up moving house, if you don't get on with some of your house-mates, or if your lifestyles are incompatible (maybe they like to party and you don't so much, or vice versa), or if they have irritating habits.
And that brings me to another issue that might be a problem, as well as there being a mismatch between your personalities and lifestyles in terms of sociability, and that's lifestyles in terms of tidiness and cleanliness. If you're an Aspie with OCD tendencies, then you might struggle to cope and you might get into arguments (or you might just be quietly upset and angry) if the people you're sharing with are untidy and leave dirty pots in the kitchen and clutter everywhere, so that's something to bear in mind as well, if you're looking for a room in a house-share, or then again you might be someone who has other Aspie obsessions and doesn't notice if the dirty laundry and pots pile up?
Another alternative, if you have major problems with looking after yourself, is supported housing. Tbh, I don't know very much about that, but I think some other people on Wrong Planet lived in supported housing, so maybe if you start a thread asking 'What's it like living in supported housing?' or something like that, then maybe other people can tell you more. I don't know what the criteria is or how long you'd have to wait for a place though, but depending on how badly you're affected by AS, it might be an option for you.