Not anymore. You either buy it through interest-free financing (often with $0 down) and have up to 2 years to pay it off, or you buy it outright at full price.
Financing it is just like a contract except that the monthly price of the phone is separate from the monthly service charge, so service with them is cheaper than the other carriers that include the cost of the phone in the monthly service charges. You can also pay the phone off early and upgrade to a new phone when you want or end service with them and switch carriers (it ends up being equivalent to a contract ETF). After it's paid off, your bill will go down if you don't upgrade to a new phone.
Verizon, AT&T, and Sprint also offer similar financing plans with Edge, Next, and Easy Pay respectively. T-Mobile calls theirs Jump. Verizon, AT&T, and T-Mobile still have contracts if you choose to go that route though.
OP: Walmart's Straight Talk has a $45 per month option. You have to buy the phone outright though.
Walmart also sells the iPhone 5s and 5c on two year contracts with some of the major carriers (like Verizon, T-Mobile, and AT&T). The 5s is $79, and the 5c is $0.97. Those are probably the cheapest new iPhones. I'm not sure watch the catch is. You'll have to ask at your local store.