Is it possible to settle the Boston Airport?

No you cannot, at least not easily. You can build stuff in it but you are limited as to what. Some of things that cannot be built are water purifiers, crops, guard posts, stores, scavenging stations, or artillery. As you said, you cannot build a settlement recruitment beacon either.

You can as noted in the comments, send settlers here from other settlements, but it would be difficult to keep them happy without being able to build the necessities.

Additional Information about this location and its restrictions:

If you side with the Brotherhood, you build the teleporter to teleport into the Institute in this location as part of the Molecular Level quest. It is also used to construct parts for the Liberty Reprimed quest.

Source: Experience and the Fallout Wiki.