Is it possible to create an app that can only be closed after you type the correct password? I want to let people use an app on my company's iPad, but I don't want them to be able to do other things with it. So they may not start other apps, et cetera.

no but you can remove all other apps including safari and add restrictions via iphone configuration utility.

They can then exit the app but really cant do much else other than load the app agian.