Given this input file (I added - ellers since you said in a comment you need to handle them too):
$ cat file
Kultur- og idrettsavdelinga skapar- eller nyska- pande kunst og utvik- lar- eller samfunnet
here's the common sed idiomatic approach:
$ sed 's/a/aA/g; s/- og/aB/g; s/- eller/aC/g; s/- //g; s/aC/- eller/g; s/aB/- og/g; s/aA/a/g' file
Kultur- og idrettsavdelinga skapar- eller nyskapande kunst og utviklar- eller samfunnet
The above works by turning all as (or whatever other char you like that's not in your target strings) into aA so we can then turn the strings we're interested in, - og and - eller, into a<some other character>, e.g. aB and aC and at that point we know the only occurrences of aB and aC in the input are the newly transformed - og and - eller since all of the existing as are now aA.
Now we can just remove all remaining -s from the file and then convert the aCs back to - eller and aBs back to - ogs and finally all aAs back to the original as.