If all the stock is sold off in a particular company, what happens to that company?

if i am understanding your question correctly, you are asking what would happen if a everyone who owned shares of a company suddenly went out and sold them all off at the same time...or roughly the equivalent.

The generic answer is that if there are no buyers of the stock, than there will be no sellers. You may want to sell it, but if no one is buying than you are stuck with those shares. Now a more specific answer, which does not preclude the first answer I gave but merely shows what will be happening behind the scenes, is this:
If the Floor Broker is getting nothing but sell orders with no buyers, than he has the authority to buy some of those shares into the Company's treasury stock. IF the funds for the buyback run dry, and NO ONE is buying, then the FP could try to sell the shares to the actual Exchange the Company is listed under. There is a representative on the floor of the exchange that could buy these shares, but only if it is to get the log-jam cleared. If this company is going bankrupt, the Exchange will not buy it and therefore they will not buy those shares.

When all is said and done, this is why the buyers of the sold shares dries up and the value of the shares you are holding dry up as well. This is what happens when a company goes bankrupt. Now, the shareholders a entitled to any left over monies, but generally they get squat, as they are literally the last ones on the totem pole getting anything.

Hope this answer helped.

usually the company has unlimited shares to issue, if they want....this may seem scary, but it is more important for a company to have a solid share value, than to float unlimited stock.

What do you mean by "sold off"? When a company first issues stock, it is all "sold off" to investors. Thereafter, the stock trades on the open market.

the only way a stock price would go to absolutely zero, is if the company is out of business. I mean completely out of business, like no assets.

Unfortunately, your question isn't clear. Anytime that stock is sold, there must be a buyer. There are no sellers when there are no buyers. Are you referring to a "selloff" as a massive decline in stock value or are your referring to a corporate buyout?

The term "selloff" is sometimes used to mean a large and rapid decrease in a company's stock price. It simply means that investors dislike the company's future prospects at the old stock price and sold it to other investors who thought that the stock was a good deal at a lower price. There are various reasons why this can happen. For example, the company may have just hit some temporary hard times, it could be in serious legal or financial trouble, or there may be an accounting scandal.

Or do you mean that all of the stock is literally sold off as in a corporate acquisition? In that case, the acquiring company purchases the stock of the acquired company from the existing shareholders. The existing shareholders may receive cash, shares in the acquiring company, or a combination of the two. The acquired company simply has new owners.