A single word for an inanimate object that is very old and held in high esteem

Solution 1:

The government building is an historic landmark.

landmark : 2. A notable location with historical, cultural, or geographical significance

Solution 2:

If it were smaller, I might well go with an heirloom, but probably not for a building. There you might may have to stick with some sort of historical monument or even a national treasure. You might also be able to work in heritage there somehow, as in a world heritage site.

Solution 3:

Currently I hear a lot of the expression heritage structure.

Though dictionaries generally define a monument as a commemorative, the term is also used in the cultural context for a structure of heritage value:

"• Monumental heritage of exceptional cultural value;" (UNESCO: Historic Urban Landscape)

Wikipedia includes the 'heritage' sense of usage:

Monument
... which has become important to a social group as a part of their remembrance of historic times or cultural heritage, or simply as an example of historic architecture.

Oxford Dict.

monument a building, structure, or site that is of historical importance or interest

I feel the example sentence sounds better with a noun instead.

The government building is a heritage monument. Many people flock to visit it every year.