.NET Core seems to be compatible with Oracle Enterprise Linux 7(OEL 7), but how about OEL 6 and 5?
I found it strange, because I thought, that OEL6 is a clone of Red Hat Enterprise Linux 6 (RHEL 6).
RHEL 6 is compatible with .Net Core, so why not the OEL 6? Is it just something Oracle based that would not work, like Oracle DB access or is there more to it?
Where did you get the idea that .NET core does not work on Oracle Linux? Here's an article where it says that you can install .NET core on Oracle Linux:
https://www.microsoft.com/net/download/linux-package-manager/centos/sdk-2.1.4
The link does specify the package created for RHEL and RHEL7 at that, but there's no mention that .NET core will not work on Oracle Linux 6. (Also, as you mentioned, Oracle Linux 6 is a clone of RHEL 6 with some minor differences.)
Now, on the question of accessing Oracle DB, you may find that issue with .NET core on any Linux or Windows machine. As of now, the only Oracle maintained provider for .NET core is in beta: http://www.oracle.com/technetwork/topics/dotnet/downloads/odpnetcorebeta-4077982.html
Additionally, the Entity Framework Core for Oracle databases is a paid package from Devart. The good news is that the .NET core team is quite active and there are definite plans on including Oracle provider for EF core in later versions.