Does the Sigma 70-300mm fit the sony A3000?

Not directly. DSLR lenses have a longer back-focus, so you need an adapter to move it forward far enough to focus on infinity (manufacturers generally make the mounts incompatible to prevent damage).

If you have a tiny budget, paying for a lens and then the $200 LA-EA1 adapter to fit it to your camera makes little sense, so buying a dedicated E-mount lens is actually better value for money.

There are third-party A to E-mount adapters, but many cut the price by leaving out aperture control and autofocus.

The Opteka is a 500mm mirror lens without autofocus, and a fixed aperture of f8 - focusing it on your A3000 will be virtually impossible. Adding the 2x teleconverter will reduce the aperture to f16. You can have good or cheap, but not both. I'd see if you can find a used E-mount telephoto lens.

The cheapo 70-300mm lenses tend to get soft beyond 200mm,

http://slrgear.com/reviews/zproducts/sigma70-300f4...

so they are a waste of money generally.

If the lens is for an Alpha mount, putting it on an E-mount requires an adapter. Generally a nuisance. Even if an Alpha to E mount adapter has better compatibility than cross brand adapters, the combo is super bulky. When choosing between a DSLR and a compact mirrorless system, it is best to realize in advance that it is a choice between compactness and extended capability.

If you want to put a telephoto on your E-mount camera, this is the one to get

http://www.bhphotovideo.com/c/product/1022364-REG/...

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