It has been around/about 40 years

Our house has been around about 40 years.

Or

Our house has been around 40 years.

Which is correct, grammatically?

I don't really see either of them as being great options. I really want to add "for" to both versions. I'm changing the example, though, because I think that the optimal phrasing for that example is something else entirely:

The concept has been around for about 30 years.
The concept has been around for 30 years.

In this case, "has been around" means "has existed". So, if we make that substitution:

The concept has existed [for] about 30 years.

This would probably pass as acceptable but I'd still probably prefer to add "for".

The concept has existed [for] 30 years.

The only difference here is that you're being precise in the amount of time. In the first version, "about" means "in the vicinity of" or "around" - which would be confusing if used in both senses:

The concept has been around [for] around 30 years.

But it's still correct, if messy.

Without "about", you're generally implying that it's been exactly 30 years. Not 29 or 32... 30. This can be a bit open to interpretation but I argue that without "about" the range of years is more confined than with it.

That said, without "for" it's more conversational, not necessarily incorrect.

When you're talking about a house, you're better served saying something more like

Our house was built about 40 years ago.