Of course you can start your own business, I know people that started their own services and started small. Their businesses didn't grow big, but since then have established firm customer base and are doing ok.
I don't know the legal details, but my guess is they have registered their services as companies.
At least as suggested by a tax preparation consultant that I went to see on an unrelated matter, you can register a business in your own name regardless of your immigration status as long as you can identify yourself and be registered in some form with IRS that they should expect taxes paid from you.
Now, what will that business be in this economy with general populace seeing their budgets shrinking and prices rising, that is up to you to decide...