Even if you could buy your own place for $6000, you need to consider that there will be new monthly expenses that you are not paying now. Right now expenses like property tax, heating and electricity are probably covered in your monthly rent. But if you bought a place you would now have to pay for these along with TV, internet, upkeep, etc.
So as others have suggested, renting is likely to be your best choice, and frankly is probably your only choice right now.
Your other post asking about lending places like Moneymart indicates that you probably do not have good spending and budgeting habits. Those lending places take advantage of people who need money quickly and who not have good credit or a bank account. You should NOT be thinking about borrowing money from lenders like these. They charge you outrageous fees for borrowing and leave you worse off.
I strongly suggest you talk to a case worker, start by going into the office where you arrange your welfare and tell them you want to discuss your financial situation and some goals you have. You need to understand how much income you have and how much you spend each month and what you spend it on. Having your own place might be possible some day but you need to start with some basics first and move gradually to your larger goals.