If you mean to store a conversation from chat locally on your own computer, then you can use whatever you would do with some other website. (Copy-paste text, print as pdf, save as html, etc.) As far as I know, there is (so far) no special feature to export some chat messages.
If you want some kind of bookmark to some conversation in chat, than this is exactly what chat bookmarks are for.
Details can be found, for example, here:
What you do is that you either click "bookmark a conversation" button in the transcript.
Or if you are present in a chatroom then you can find this in the dropdown menu on the right.
Then you simply follow the instructions. You select the first message of the conversation. Then you move to find the last message you want to be part of your conversation and click on it.
What you achieve by this is that the particular conversation appears in two places. One is conversation tab of that specific room. (Here is the conversation tab of Mathematics chatroom.) And it also appears in your chat profile among conversations. (I will use my own profile as an example.)
Some useful things to keep in mind:
- The conversation tab is a shared resource for all users of the chatroom. So you should keep this in mind and not use all space there for yourself. (But if you do not bookmark something too often, you're probably fine. And, of course, this is a non-issue for a room which is used only by a few users.)
- You can also remove the conversation. If you delete a conversation and then create a conversation in the same room with exactly the same name, the url will also be the same. (This is useful if you want to keep the same link - for example if you already linked to the conversation from elsewhere - but you want to add more messages.)
- If you post a link to conversation as a chat message, it is oneboxed.