Hello Vanessa,
For a property to actually "pay for itself" you will need to find a virtually impossible combination of circumstances which almost never come together:
1. Very low interest rates - this insures that you pay the least possible to the bank in interest for your purchase
2-3. High yield/low purchase price. These two things go together. The yield is the % of the purchase price which is paid to back to you each year by the rental income. A typical yield would be, say, about 4.5%. You would need a high rent and a low purchase price to meet your criteria.

For a property to pay for itself, the interest payments for the loan secured to buy the property would have to be totally covered by the rental income.

First of all you must realize that banks (almost) never lend you 100% of the purchase price.

Theoretically, in order for a property to pay for itself, you would have to secure a loan at a rate exactly equal to the yield rate. For example, you get a loan at 5% and you enjoy a 5% yield on your holdings. In reality even this doesn't actually work because bank loan rates are calculated in a way which is systematically disadvantageous for the mortgage owner. (interest is paid back in abundance first and principal is reimbursed progressively towards the end of the loan period). So in practice, a loan at 5 % could not be paid back with a yield of 5%.
In short, finding such a property is for all practical purposes virtually impossible.

I currently have a property in France whose mortgage is being paid by the rental income. This said, it was bought 12 years ago and, until this year, we lived in the property. The prices in 12 years have nearly tripled, and the mortgage now has only three more years to run...

I hope that this helps.

Neustria