Four private astronauts launched to orbit by Elon Musk’s SpaceX returned to Earth Saturday evening, splashing down into the ocean off the east coast of Florida after a three-day mission.

A company space capsule ferrying the quartet landed under parachutes at 7:06 p.m. ET. The capsule carried the ship’s commander, billionaire entrepreneur Jared Isaacman, as well as Hayley Arceneaux, a physician assistant; geoscientist Dr. Sian Proctor; and Chris Sembroski, an aerospace data engineer.

Mr. Isaacman was heard saying shortly after touchdown that it was a “heck of a ride” and that things were “just getting started.” Boats moved toward the capsule following the landing, according to the live stream.

Boats moved toward the capsule following the landing, according to SpaceX’s live stream of the return. Roughly 30 minutes after landing, teams had hoisted the spacecraft out of the water and onto a recovery vessel. Crew members began leaving the vehicle through a side hatch and onto a platform around 50 minutes after splashdown, with Ms. Arceneaux emerging first.

Medical personnel were expected to check on the crew, who would then be flown back to land, according to the livestream.

In a tweet Saturday evening, Mr. Musk congratulated the mission.

The Inspiration4 mission broke new ground for a private human space flight, marking the first occasion that a crew of people who weren’t government astronauts traveled to orbit on a company’s spaceship. Previous tourists who traveled that deep into space purchased seats on Russian government rockets.

The mission marked other firsts as well. Ms. Arceneaux, who is 29, was the youngest American and woman to travel to space, according to a spokesman. Dr. Proctor, designated as the pilot for the trip, was the first Black woman to serve in that role.

SpaceX took the crew to the deepest orbit that Americans have been to since 2009, when the National Aeronautics and Space Administration last repaired the Hubble Space Telescope.

Benji Reed, senior director of human spaceflight programs at SpaceX, described the flight as a smooth one during a briefing Saturday night after the landing. The company saw issues with a waste-management system and took a temperature sensor on an engine offline because it was providing bad data, he said. That sensor had backup ones in place, and the crew never faced any risks, he said.

The issue with the waste-management system had to do with a fan and workarounds for the problem were found, Inspiration4 mission director Todd Ericson said.

Orbital space missions can pose risks. The crew orbited Earth at around 17,000 miles an hour in SpaceX’s Crew Dragon capsule. While re-entering the atmosphere, the vehicle’s heat shield was expected to face temperatures of around 3,000 degrees Fahrenheit.

During the flight, Mr. Isaacman and Dr. Proctor were tasked with ensuring the automated system that flew the Crew Dragon capsule was working properly and were prepared for any contingencies, Mr. Ericson said.

Mr. Musk’s company launched the crew toward orbit Wednesday evening. The mission used a space vehicle SpaceX has also tapped to take NASA astronauts to the International Space Station: a Crew Dragon capsule stacked on top of one of its Falcon 9 rockets.

During their trip to orbit with SpaceX, crew members held video calls with children at St. Jude Children’s Research Hospital, the recipient of its charitable funds; conducted scientific experiments; and floated around while taking in views of Earth. On Friday, they appeared on video at the New York Stock Exchange at the close of trading. Meals on board included pasta Bolognese, lamb and cold pizza.

“Sorry it was cold! Dragon will have a food warmer & free wifi next time,” Mr. Musk said in a tweet about the latter item.

Private space trips are one part of a commercial space industry that also includes rocket-launch startups and venture-backed companies focused on refueling satellites, as well as larger aerospace companies with decades of experience building space vehicles for government clients.

SpaceX isn’t alone in seeing opportunity in tourist visits to space. This summer, Virgin Galactic Holdings Inc., founded by Richard Branson, and Jeff Bezos’ Blue Origin LLC completed high-profile trips to the edge of space with their founders on board. Boeing Co. has said seats on future missions on its Starliner capsule, which has yet to carry people to space, could be bought by tourists.

Executives have said SpaceX has a backlog of commercial trips booked and that it wants to conduct at least six such missions a year. Mr. Reed, the company’s senior director of human spaceflight programs, said Saturday that interest in trips had been growing.

It might take years before more people are able to access space, in part because of the costs of such trips and relatively limited capacity. Virgin Galactic said in August it would charge consumers at least $450,000 a seat for its flights, while Blue Origin sold for about $30 million a seat on its rocket that took Mr. Bezos to space.

Space tourism might generate close to $4 billion in annual revenue by 2030, according to an estimate from UBS. Access to space will broaden when the equivalent of plane loads of people are able to travel there, said Myles Walton, a UBS analyst.

Mr. Isaacman, the chief executive of Shift4 Payments Inc. who paid for Inspiration4, hasn’t said how much he paid for the trip but has said it was less than the amount the mission aims to raise for charity, which is $200 million.

The mission is a “very, very small steppingstone in the bigger dream of making space accessible to everyone,” Mr. Isaacman said in an interview before the takeoff.

Write to Micah Maidenberg at micah.maidenberg@wsj.com