Technically, of course. Almost any company can go bankrupt. One small note: a company goes bankrupt, not its stock. Its stock may become worthless in bankruptcy, but a stock disappearing or being delisted doesn't necessarily mean the company went bankrupt. Bankruptcy has implications for a company's debt as well, so it applies to more than just its stock.
I don't know of any historical instances where this has happened, but presumably, the warning signs of bankruptcy would be evident enough that a few things could happen.
Another company, e.g. another exchange, holding firm, etc. could buy out the exchange that's facing financial difficulty, and the companies traded on it would transfer to the new company that's formed. If another exchange bought out the struggling exchange, the shares of the latter could transfer to the former. This is an attractive option because exchanges possess a great deal of infrastructure already in place. Depending on the country, this could face regulatory scrutiny however.
Other firms or governments could bail out the exchange if no one presented a buyout offer. The likelihood of this occurring depends on several factors, e.g. political will, the government(s) in question, etc.
For a smaller exchange, the exchange could close all open positions at a set price. This is exactly what happened with the Hong Kong Mercantile Exchange (HKMex) that MSalters mentioned. When the exchange collapsed in May 2013, it closed all open positions for their price on the Thursday before the shutdown date.
I don't know if a stock exchange would simply close all open positions at a set price, since equity technically exists in perpetuity regardless of the shutdown of an exchange, while many derivatives have an expiration date. Furthermore, this might not be a feasible option for a large exchange. For example, the Chicago Mercantile Exchange lists thousands of products and manages hundreds of millions of transactions, so closing all open positions could be a significant undertaking.
If none of the above options were available, I presume companies listed on the exchange would actively move to other, more financially stable exchanges. These companies wouldn't simply go bankrupt. Contracts can always be listed on other exchanges as well.
Considering the high level of mergers and acquisitions, both unsuccessful and successful, in the market for exchanges in recent years, I would assume that option 1 would be the most likely (see the NYSE Euronext/Deutsche Börse merger talks and the NYSE Euronext/ICE merger that's currently in progress), but for smaller exchanges, there is the recent historical precedent of the HKMex that speaks to #3.
Also, the above answer really only applies to publicly traded stock exchanges, and not all stock exchanges are publicly-held entities. For example, the Shanghai Stock Exchange is a quasi-governmental organization, so I presume option 2 would apply because it already receives government backing. Its bankruptcy would mean something occurred for the government to withdraw its backing or that it became public, and a discussion of those events occurring in the future is pure speculation.