Chickens are soulful animals, Phil Tompkins says. “They walk up to you and talk to you,” he says. “They want to interact with you. They’re like a cat or dog, except they’re a chicken.”

That vision of affability, along with the animal’s more obvious role in the food chain, are among the reasons that Mr. Tompkins and his wife, Jenn, who describe themselves as homesteaders living in western Pennsylvania, decided earlier this year to start a business called Rent the Chicken. For $350, the couple will deliver to customers a pair of egg-laying hens, a coop, a supply of food and a water dish for a rental period typically lasting from May through November.

Rent the Chicken is one of a handful of backyard-chicken rental businesses that have cropped up in states around the country, including Michigan, Massachusetts and Maryland. They join what is known as the sharing economy, a growing category now estimated at $3.5 billion that includes companies offering products or services without transferring ownership.

Most such businesses fall into two categories. There are so-called peer-to-peer businesses, like Airbnb, RelayRides and SnapGoods, which allow people to rent out their houses, cars or just about any other possession they might not be using for a short time. And there are those like Rent the Chicken and the couture-rental service Rent the Runway that follow a more traditional model in which a business owner buys goods and rents them out.