No its not. For a object to actually become a sun, takes a large enough mass to ignite the thermonuclear fusion at its core required for it to actually *be* a sun. There are objects in the universe known as "brown dwarfs" which are essentially failed stars, as they don't have the mass required to sustain the fusion. They are essentially giant gas planets, many times the size of Jupiter.
A "mini-star" that could orbit the earth is impossible as far as we know. While stars can be that size (white dwarfs) their density would be very high, with a mass the same as the sun scrunched into a volume the size of the earth. We'd still orbit around that sun in that case due to the gravitational pull, and since white dwarfs are the product of a dying sun, we probably wouldn't be here anyway.
Edit: For reference, this is the actual size of the sun compared to earth: http://didyouknow.org/graphics/space/planetsizes.jpg
And the sun is a comparably small star: http://2.bp.blogspot.com/_RsCCizsWvNw/SU7dwO8JYBI/AAAAAAAABQ0/BlEh-2hgKBw/s400/biggest+planets.jpg