If I construct an island in the middle of the Pacific ocean (not within any country's water territories) could I claim sovereignty?
This is covered by the UN Convention on the Law of the Sea. You specify that the island is constructed "not within any country's water territories": this is the "high seas", covered by Part VII of the treaty. Specifically relevant to you is Article 89:
No State may validly purport to subject any part of the high seas to its sovereignty.
Yes, you can construct an artificial island (Article 87, part 1d), but you can't claim sovereignty over it.