Yes, it is possible. There are organisations who act as third-party sponsors for J-1 visas. An example is http://cipusa.org/. These organisations help with some of the paperwork, lowering the amount of work required by the applicant (you) and the company.

You are probably right in that the amount of formalia and paperwork is rather large for a small startup, but there are certainly larger companies that are willing to put in the work. I did an internship at a company in the US, and I live in Sweden.

I urge you to try, an internship in another country can be a great learning experience.