What is this structure? Can I infiltrate it?

Solution 1:

That is what's known as a Set Piece.

There are several set pieces in the game; some useful, some not so useful. The one you're looking at, which has been called the "Insane Pig" by some members of the community, is absolutely worthless.

There seems to be nothing special about the pig or home within the indestructible Basalt. They act just like a normal pig and home anywhere else on your world, producing the same items upon death/destruction as the others.

As for the Obelisks, well, you can infiltrate those (and it's probably why it got the name "insane pig"). In order to lower the obelisks, allowing you to walk across them, you must lower your sanity to ~15 or lower. So, it's really up to you if you think that's worth it.

Solution 2:

Per the Wiki and above comments:

Basalt cannot be destroyed (Maxwell says he created a rock that is too heavy for even him to lift). Obelisks cannot be destroyed.

Obelisks can be lowered if you lower your sanity enough (by eating green mushrooms).

The pig house remains a mystery, though. I haven't had one spawn in this way. In the picture there is a regular pig up there, so it's unlikely to be the home of a Pig King or anything. And, the fact that there is a real pig there makes it unlikely to be a trap.

My WAG (Wild Assed Guess) is that it is nothing more than a regular pig house. This must be one of the random structure spawns that the developers have put in. Much like the others, there is little rhyme or reason to the way they've spawned, except that they represent structure types. (e.g. A wooden fortress filled with beefalo wool despite there not being a beefalo in the area ...)