Google Hangouts, by default, keeps a neverending chat log in Google's cloud, but there is actually a way to delete that log. The log in question, as long as you've never interfered with it before, will have literally every message you've ever sent over Hangouts, and can become a source of clutter or other trouble. This makes it all the more relieving that getting rid of all of those old messages is quite simple. It's worth noting that this only applies to your devices and account; other people involved in the old messages will still have them. Another caveat is that it can't be done from a mobile device, though if you don't have a desktop computer handy, you can always use a public computer or borrow a friend's laptop and use Incognito Mode or an equivalent.
Once you have a keyboard under your fingers, head to your Gmail page or hangouts.google.com, where you'll find the Hangouts web app. If you've downloaded the native app for Windows or Mac, you can do this there, as well. Simply right-click a conversation you want to delete, and click on "Leave". You can also archive conversations, if you so desire. You can archive conversations in the mobile app, though you can't leave them. To keep logging from happening, you'll want to go into a conversation, click on the gear icon, then uncheck "Conversation History". This will create messages that can only be seen briefly on your device. Note that the other person may still be able to keep the messages, if they haven't taken this step. It's also worth noting that Hangouts lacks end-to-end encryption, so even with this turned on, anybody on the same network as you or with access to your account could potentially dig messages up.
Hangouts is in a bit of a strange place in its life cycle right now, with Google pushing consumers toward its other options in subtle ways without killing off the service entirely, while on the enterprise side of things, Hangouts is being diversified and strengthened. Expect the current feature set to be how the app will operate indefinitely on the consumer side, unless something changes.