I am self employed, 68 years old and was surprised that I had to pay 15% social security tax on my small earnings.
My question is simple - is this tax on my my gross earnings or my net?

Gross is taxed.

You will report yourself employed income on schedule C. On sch. C, you can deduct your business related expenses. The net income (after deducting the expenses) is subject to SE tax at 15.3%.
When you are self employed you pay social security and Medicare for employee as well as employer. When you are employee you pay only your own part that is 7.65%.

For more on self employed income and filing, read Your U.S. Tax Return: Tax Filing by Self Employed Sole Proprietor or Independent Contractor