It is possible for owner's equity to be a negative amount. The following illustrates how it might occur.
In 2008, a sole proprietorship was begun with the owner investing $100,000. During the years 2008 through 2011 the owner withdrew most of each year's net income. As a result, the total owner's equity at the end of 2011 was $110,000 (original investment of $100,000 plus $10,000 of net income not withdrawn). During 2012 the company's expenses exceed revenues by $125,000 and there were no draws or additional investments by the owner. The owner's equity at the end of 2012 would be a negative $15,000.
The negative amount of owner's equity also means that the company's balance sheet will report liability amounts greater than the amount of assets. The company could operate under those conditions if its assets are turning to cash before the liabilities need to be paid.