Translation:That story is funny!
In Chinese it's rare that and adjective isn't preceeded by 很 so it can be interpreted as either English sentence.
Not “rare” in the general sense. You need to know when it would sound unnatural without 很, which is hard to elaborate completely.
很 in this sentence is function word used for sounds smooth only. Actually, it's more close to verb "is". In many adjetive, you can see the same usage.
For example: 他很棒 He is great. If you wanna describe very, then say 他非常棒. Or we emphasize 很(make the long sounds) when speak, it also means very.
Given how most questions here will translate "很“ to "very" it maybe should be included in this question as well even though it is unnecessary.
Gramatically, it's right. But in chinese concept it's different description for different circumstances.
If you say that's a funny story, it emphasizes on story. The translation will be 那是個好笑的故事. It's just an expression, and we feel the topic ends now.
In this sentence is talking about funny. And usually following you will talk about why you think it's funny. So it's the beginning of the conversation.
I know it doesn't make sense, but this is cultural gap. We talk with many implications.