Your post is not clear on what you mean by "without a firewall". There are so many places in a typical setup where a firewall could be placed, and yes, it is safe to leave them out in some situations.
For example, your store has a firewall at the internet connection and everything inside is a private ip address. The cash registers run on their own network, firewall'd away from the other computers in the store, with rules to allow for outgoing credit card authorizations and that sort of thing. Does each cash register need a firewall? Probably not, and it might even be a significant expense to maintain updated rules every time the network needs to be reconfigured.
So yeah, it depends on the entire configuration. The tone of your post suggests that the situation is not good though, and of course, it's a lot easier to argue for a firewall these days than against one.