It's great your into sailing and wanting to live aboard a boat at your age. I'm young (25) like you and I hardly ever run into people my age that have an appreciation for sailing like I do so its nice to hear from someone else.
I would say if you can swing it on that budget than definetly do it - it will be a great experience and you will learn a ton about boats in general. I don't know how much boats are going to cost in your neck of the woods and how big and in what condition you are looking for, but it may be tough to find one big enough for a live aboard and in descent condition. I started off with a small 25' sloop that I lived on every weekend and a couple weeknights (other nights I spent with my gf). When I bought the boat I thought it would be perfect as it was in pristine shape and as my first boat it felt plenty big to me at first. However I wound up only keeping it a year before trading up to a 32' for more room and accomodations (head, real stove, plumbing, electric, standing head room, radar, shower, etc, etc.) In that one year I found out that I absolutely loved sailing and being on the boat but the 25' just felt more like camping out as oppossed to living on a boat.
Now I love the 32' and feel so lucky to be able to have it and enjoy it so much but the downside is it was much more expensive. You can probably find a 25'-27' boat in pretty good sailing and live aboard condition for under $6k, but it could get real small, real quick. Now if you move up to a larger boat say around 30' or more the problem is the cost goes up exponetionally quick because instead of having real basic electric and plumbing, they have much more complex systems and more of them. So you may find a larger boat for under $6k but unless it is an absolute steal it probably won't be in the best shape.
One big point is that a lot of this stuff is very subjective-both the condition and the size of boats. As for myself personally, I am a neat freak and I only looked at and currently keep my boat in pristine condition and would not look at boats that were messy and in poor shape. However there are plenty of people who have different opinions and don't mind living aboard a wreck - to them they are just happy having the boat and that is abolutely fine. The same goes for size. I am 6'3 and needed a bigger boat because for what I used the boat for (mainly sailing and sleeping and living aboard) 25' was too small. For others 25' is perfectly fine.
So in all to sum things up I think you can get by with a smallish boat in your budget in pretty good shape or you can one slightly larger that needs a lot of work but will give you more space all depending on what you can tolerate as a person. Once you have the boat you will find that they are also a lot of money just for upkeep and that goes for any boat even ones in perfect condition brand new off the showroom floor so the montly budget may also be tight. Between moorage, maintenance and upgrades I probably spend $8-$12k per year so it does get costly. With all that said though, there is nothing better than sitting in the cockpit of your own boat watching the sunset knowing she is all yours.
As a matter of fact, I'm getting ready to head down to the marina in about an hour to go spend the night on mine for the Memorial Day weekend and even now after 3 years of owning her I still get goosebumps and extremely excited at the thought of staying aboard and just being on the water and to do it at such a young age is great.
Sorry for such a long post but I guess I feel I can relate to being young and wanting to get and live on a boat more than anything else in the world. For me the dream came true so hopefully it does for you as well. Best of luck!