So, I've been watching Warehouse 13, and I've seen at least 3 episodes with actors from Eureka in the show. Of these, the first 2 had the same actors with different people, the last had the same actor acting as the same person. What this an oversight, or is there some in-universe explanation for why the previous actors were acting differently?
Eureka, Warehouse 13, and apparently Alphas, all exist in the same universe. This is supported by a number of cross-over episodes, namely Warehouse 13 S2E5: 13.1, S3E6: Don't Hate the Player, and Eureka S4E5: Crossing Over. Each episode makes explicit reference to established facts in the other series. From those three episodes, it's clear that Warehouse 13 and Eureka inhabit the same universe. The proof for Alphas existing in the same universe is much less convincing. In S1E5: Never Let Me Go, Lindsay Wagner reprises her role from Warehouse 13 as Dr. Vanessa Calder, although no mention of the Warehouse is made (albeit for the obvious reason that it's a secret). On a side note, a huge opportunity to tie Alphas to Eureka was missed in Alphas S1E7: Catch and Release, but that's not important.
In at least 2 different episodes of Warehouse 13, there are references that the Warehouse is "not Eureka" or that some aspect belongs in Eureka, not the Warehouse.
As for actors in Eureka or Warehouse 13 appearing in the other series as another character, that's because there's a limited amount of acting talent in Vancouver, and Eureka and Warehouse 13 use the same casting directors, so when they see someone they like for a minor role, they're probably going to call them back before a casting call. This isn't the first time this has happened either: Star Trek (Jeffrey Combs appeared in 3 different series as 3 different characters. Marc Alaimo, best known for his portrayal as Gul Dukat in DS9, not only played a different Cardassian in TNG, but 3 other characters in TNG as well. Other examples abound.), Stargate (Brandy Ledford played a Tok'Ra in SG1 and a planet-of-the-week inhabitant in Atlantis as an example, there are many more), the list goes on. This happens a lot in TV, and it's not restricted to sci-fi. There are examples from many other genres as well.
In universe: They're different characters who just happen to look alike.
Out of universe: Warehouse 13 and Eureka are both produced by the same company, and perhaps there's some synergy going on with casting. In addition, sci-fi shows often share actors, with actors from one sci-fi show working on other sci-fi shows (e.g. Summer Glau). This is even true within franchises. Some of the cast members in Stargate:Atlantis and Stargate:Universe played different characters in minor, one-episode roles in SG-1, in a fashion similar to the W13/Eureka actors in your question.
I don't think there's a real "in-universe" explanation.
Out-of-universe, it's a common scenario in Sci-Fi to have actors appear in other series - aside from Keen-mentioned StarGate crossovers, there's the (in)famous "Jason X"/"Andromeda" one.
In "Jason X", Lexa Doig's Dr. Rowan had an android KM14, played by Lisa Ryder - which was a (likely intentional?) gag based on the fact that Lexa Doig played an android Romy in TV Sci-Fi series "Andromeda" together with Lisa Ryder's Beka Valentine character.
Global dynamics is the main research facility in eureka, where Fargo works. Warehouse 13 is another secret facility in America. Fargo came and installed some computers in the warehouse and later claudia from the warehouse came to eureka and farted about with Fargo.
There is nothing complicated about their coexistence, two government operated depts both dealing with some freaky shit.
I'll admit I was already planning on watching Warehouse 13 but after watching the Eureka episode Crossing Over, I started watching Warehouse 13 to see the actress. But shortly into the first episode, they had a person show up who I immediately recognized from Eureka. The actor is Saul Rubinek. He apparently was only in one episode of Eureka but he was interacting and "connected" to what they referred to in Eureka as "The Artifact". So when he showed up in charge of a warehouse for artifacts with powers, I figured it might have been as the same character. I had to look up the name though. Though it seems I've heard him using the name Artie before as character. I'm surprised no one else picked up on that yet.
It may not be an oversight but rather a delayed idea as not all crossovers are planned from day one
Simply put eureka and warehouse 13 weren't in the same world originally but were merged to spark cross following interest E.g eureka viewers may never watch warehouse 13 due to lack of interest till Claudia visits