Very difficult to do both simultaneously, but there are certainly techniques to do one or the other, and to come close to doing both.
The no down payment is easiest, even in today's market. FHA program and individual city/state program can effectively eliminate the down payment.
There are other purchasing techniques that will also eliminate or sharply reduce any downpayment. For example, using a lease-purchase or lease-option structure, you may be able to reduce your up-front cost to a minimal option fee. Or even no upfront option fee; the "consideration" is the value of the lease portion of the lease-option. I've done that: lease-optioned a $253,000 property for no upfront cash--just the value of a one-year lease at $1,500 a month.
You can also purchase a property "subject to" the existing mortgage. The current owner gives you the deed in return for your promise to make the mortgage payments. Sometimes, however, there are back payments due, or the owner will ask for some cash for moving expenses or other uses. Even then, though, you may be acquiring a property for 1%-2% in payments to the lender or to the owner, versus 5%, 10%, or more if you were applying for a new mortgage.
In any variation of seller financing, the downpayment, if any, is negotiable between you and the owner. They'll usually want some sort of downpayment, which is completely understandable. But it's negotiable. And I can envision circumstances in which the amount you offer--above current fair market value--perhaps coupled with a relatively short balloon payment--would eliminate the need for a downpayment.
OK, so there are a whole lot of ways to eliminate the downpayment. There are other techniques, too, in which a downpayment would be required, but not from you. Going into it with a partner, in an equity share arrangement, would be one.
Now, let's move on to buying a home with no mortgage payment. That's tougher though, as Bill Clinton said, "It depends on what the definition of 'is' is." In this case, it depends on what the definition of "mortgage" is.
Deborah was close when she said that the only way not to have a mortgage payment was to pay in cash. Actually, the only way not to have a mortgage payment is not to have a mortgage. And therein lies the difference.
Returning to the lease-option scenario, you will have a monthly payment, but the payment will be comparable to rent, not to a mortgage. (Hence, my comment about it depending on what your definition of mortgage is.) The owner of the property still has a mortgage; you don't. So, for instance, in the Northern Virginia area a house worth $500,000 might rent for $1,600. A new mortgage might cost $3,000 a month, plus taxes and insurance. But if the current owner bought the house 20 years ago for $150,000 and his total monthly payments are $1,300 including taxes and insurance, you'd probably be able to lease-option it for $1,600-$1,900 a month. You haven't gotten a new mortgage; the owner hasn't gotten a new mortgage, and your payments are, say, $1,800 a month versus $3,000.
Another no-mortgage scenario: seller-financing. In this case, technically there would be a new mortgage, but one created by the seller. And, depending on negotiations, as with the no down payment scenario above, it might be pegged at well below what a new mortgage would cost. A wrap mortgage would work the same way.
Another scenario in which you wouldn't be paying for the mortgage; a properly (for you and this scenario) equity share arrangement. Now, usually, an equity share is constructed so that an investor puts up the downpayment and is a co-borrower; the resident usually pays the monthly mortgage. But there's no reason why this must the only model. You want no mortgage? Fine. Have the investor put up the down payment and be responsible for 50% of the new mortgage. Your responsibility would be the remaining 50% of the mortgage, plus maintenance of the property. In return, at the conclusion of the equity share arrangement, the investor would be entitled to a greater share of the profits.
Or the same scenario could be accomplished through variations on land trusts. You find someone who wants to sell, but doesn't need their equity out immediately. You find an investor to supply some initial capital for the contingency fund. The investor also agrees to pay part (or most) of the monthly payment to the trust. You pay the remainder. At the conclusion of the trust, the property is brought out of the trust and disposed of according to the trust documents. Typically, the resident beneficiary would have first right to purchase at fair market value; the investor beneficiary would be second in line. But it could be structured in any way you wish.
So, there. We've figured out a number of ways to eliminate any down payment. And we've identified a number of ways to eliminate or minimize a mortgage payment. And these two techniques can work together to accomplish your goals.