The short of the long story is this: the middle and passive forms in Greek are identical and, consequently, it is impossible to determine grammatically which form (active/middle or passive) the author intended to convey.
The best that I can say it read "having been set (tetagmenoi) into life everlasting." It's ambiguous as to when and by whom, although just about in every instance in the Bible that I could find the verb τασσω (tasso), meaning to set or arrange, etc. identifies the one who is setting or arranging.
This is not the only example of ambiguous or indeterminate verses. For instance, the ἐφ᾽ ᾧ πάντες ἥμαρτον (ef ho pantes hemarton) in Romans 5:12 can be read in two different ways and, like Acts 13:48, understood in a manner that is theologically as different as night and day! One reading led to the Eastern view that the original sin was not inherited and the Western (Auguistinian) view that it was!
The Reformed argument that the middle voice is "almost never used" is weak, imo. Almost doesn't mean never. This argument is made that much weaker by the fact that "almost never" or truly never used exceptions are not dismissed in other instances.
For example, nowhere in the New Testament does the standalone ανοθην (anothen) in John 3:3 mean "again," but most English-language Bibles translate it as such—for doctrinal reasons, ignoring the lexicon completely.
Other than the Reformed "almost never used" argument as their only Trump card, other sources point to a different conclusion.
I consulted the Church Slavonic version which is the closest in my opinion to the Greek (simply because Church Slavonic was created based on Greek and for the purpose of one-on-one translation from Greek). There is no doubt that it renders Act 13:48 in the passive voice...
...but there is also no indcation of any predestination implied here, implicitly or explicitly!
This is good because there is no ambiguity in the Slavonic version whatsoever as regard their appointment bieng made right there and then.
The Greek Church never taught the Reformed-type predestination, and the Slavonic version certainly does not read it as something eternally arranged. Rather they were set (not pre-set) for salvation by believing.
In summery, the Eastern view is that "Divine determination depends on the life of a man, and not his life upon the determination." [Bishop +Theofan the Recluse]
(like the sabbath was made for man and not man for sabbath... Unfortunately, the Western Church wasn't so lucky as to stick to its original teaching. The Latin Vulgate Bible added the word prae to the original ordinati to create praeordinati (lit. pre-arranged, pre-ordianed) during St. Augustine's lifetime (4th century), obviously under his inlfuence and his strong denial of free will (before he recanted).
Prior to that the Latin text read the same as the Slavonic does, i.e. simply "they were arranged (ordinati) to life everlasting." by believing.
However, this detail is interesting to me as evidence of an ongoing corruption and manipulation of biblical manuscripts in the early Church (gasp!) in order to make the "pristine" word of God conform to the evolving doctrine!
The Bible, as usual, being so thoroughly redacted over time, is full of contrasting statements. Thus, biblical support exists for predestination (mostly Paul) and free will (mostly the Old Testament), and Act 13:48 is no exception to this dichotomy.
Finally, I found this, imo, fairy well styled orthodox summary of Acts 13:48 as given by John Wesley: