Yep, there are a couple of different options here!
First is the Food Pen, located on the top left corner of the page (it's the apple icon). You can put food here, and it will automatically feed your pets every time their hunger level gets low. You start off with a limited number of food slots, but you can add more food by upgrading your Food Pen. The more times you upgrade, the more food you can add, and the longer you can stay away before your pets go hungry.
The second option is to put your pets in your Showroom. Showrooms are primarily for showing off pets -- kind of like a pet gallery. But they're incredibly useful for storing your pets while you're away, as pets in the Showroom do not age or go hungry. This means you can leave them in there as long as you need to without them dying. Once again, to add more pet slots, you must upgrade your Showroom. Your Showroom will be located under the "Shops" tab at the top of the page.
Hope this helped!