语法批改，谢

第一句话缺少谓语动词。改为：The British Museum is one of the earliest(原句该单词有拼写错误） museums in the world.
面积是六万多平方米，应该用一般现在时，而且你的原文里“多”没有写出来。改为：It covers more than 60 thousand square meters
陈列室：showroom 或者exhibition room。后面there over 8000...一句也是缺少谓语动词。另外，一百万就是a million，八百万就是eight million。这两句话可以合并成一句，改为：It includes over 100 showrooms, with over eight million collections.
以图书馆闻名一句有错误，应该用被动态has been known by people，不过这样写给人感觉有点中式英语。建议修改为：The museum is well-known for its library, a place where...。
be well-known for sth表示以某事物著称（for这里表示原因）