Ryan Gosling and Eva Mendes's Disneyland Date

By Paul Chi

09/07/2011 at 07:00 AM EDT

Sorry, ladies: Hollywood's most eligible bachelor with abs of steel may be off the market.

Ryan Gosling enjoyed a casual date with Eva Mendes at Disneyland's California Adventure Park in Anaheim, Calif., on Saturday. The pair spent the day holding hands and getting cozy well into the night.

"They looked relaxed and seemed to be having a fun time together," an eyewitness tells PEOPLE. "Like a dating couple, they always walked very closely, holding hands and their arms touching side-by-side."

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After enjoying some pink cotton candy and popcorn, Gosling, 30, and Mendes, 37, had a blast climbing aboard the amusement park's rides like Ariel's Undersea Adventure ride and the 4D Toy Story Mania ride. The pair also watched the World of Color Spectacular water show and later spotted at Disney's Tomorrowland around 10 p.m., holding hands again.

"Ryan seemed to pay a lot of attention to her," says the onlooker. "He would glance over and look her in the face as they walked through the crowds making sure she was okay."

The two, who recently filmed the drama The Place Beyond the Pines together, are taking their onscreen romance off camera.

"Ryan and Eva have spent a lot of time together on set," says a source. "They have great chemistry and seem to enjoy filming together. They seem to be good friends and always laugh together."