It's true that there is no exemption from tax on capital gains below a certain amount (other than when selling your primary home). However, if the gain is long-term and your other income is low, it's possible that part or all of the capital gain will be taxed at a 0% rate. In that case, while the entire gain is technically taxable, there is effectively no tax on the portion that is taxed at 0%. (This might be what your neighbor's wife is thinking of, but if so, she got the details wrong.)
You don't lose the 0% rate if selling at a higher price pushes your total taxable income above the limit for the 0% rate. Only the part of the gain that falls above the limit is taxed at a higher rate. The portion that is below the limit still gets the 0% rate. The taxable income limit for the 0% tax rate depends on your filing status, but it is not $25,000 for any filing status. And the limit applies to your total taxable income, not just the gain on the sale.
