Renting was the act of making use of a space owned by another, sometimes in exchange for a fee, called rent. Rent was a fee payable to a landlord in exchange for temporarily making use of a building or unit. The document associated with renting a property was called a lease.
According to Quark, on Ferenginar, pregnancy was considered a rental. (DS9: "Nor the Battle to the Strong")
In 1893, Mrs. Carmichael stipulated that rent was payable by one PM Wednesdays. When she came to collect on Thursday, he assured her she would soon receive rent along with a bonus. Disbelieving him, she demanded her money that Friday or he and the rest of his theater troupe would be evicted. (TNG: "Time's Arrow, Part II")
In 1930, Edith Keeler mentioned to James T. Kirk and Spock that if they needed a place to stay, the place where she lived had a room for rent at a rate of $2 a week. A radio repair shop had a sign in the window indicating it was for rent. (TOS: "The City on the Edge of Forever")
In 2263 of the alternate reality, Leonard McCoy questioned Starbase Yorktown's space-based construction and wondered why they couldn't just rent some space on a planet. (Star Trek Beyond)
In 2365, Q compared the scenario he had placed Captain Jean-Luc Picard and USS Enterprise-D's crew in to a dance; "The hall is rented, the orchestra engaged. It's now time to see if you can dance." (TNG: "Q Who")
The same year, while Picard played the Dixon Hill series holoprogram, Madeline informed "Dixon Hill" that his landlord wanted the rent, or he would be evicted from his office. (TNG: "Manhunt")
Deep Space 9's Promenade had several spaces for rent during the 2370s. (DS9: "Destiny", "The Die is Cast")
In 2371, Quark wanted to rent the shop beside his, which was going out of business, for Cardassian gaming concessions. Both Odo and Benjamin Sisko assured him that no live vole fighting was to take place on the Promenade. (DS9: "Destiny")
Later that year, attempting to convince Elim Garak to maintain his tailor shop, Odo told him that Quark was considering renting it and putting in an Argelian massage facility. Both Garak and Odo agreed that Sisko likely wouldn't stand for that. (DS9: "The Die is Cast")
In 2372, Quark found the Federation to be ideal landlords, as they demanded no rent, repair fees, or power fees from him for his operation of Quark's. Wanting the worker's strike to end peaceably, Benjamin Sisko then gave Quark an ultimatum; pay five years of back rent and other fees – a considerable amount of latinum – or talk with Rom and the other strikers. Quark chose the latter. (DS9: "Bar Association")
In 2375, annoyed by Nog's persistent playing of the song "I'll Be Seeing You", Jake Sisko told him that if he wanted to hear that song again, he should rent a holosuite. (DS9: "It's Only a Paper Moon")