One extrapolation is Modern Greek, which (as Rich Alderson’s answer says) has them as short mid-high tense: [e̞ o̞].
Sidney Allen’s Vox Graeca is the authoritative work in English on Ancient Greek pronunciation and the evidence we have for it, and it treats short mid-high tense as the default assumption. It rejects the notion that they were close mid [e o], because that would have likely clashed with the newly monophthongised <ei> <ou>, which were pronounced [eː oː]. He concedes that <ei> [eː], which historically reflected Homeric <ee>, was the long version of /e/;
But phonetically Attic ε probably lies midway between classical η [ɛː] and ει [eː], and there seems nothing to be gained by setting it in a special relationship with either.
I must say that I don’t find this argument convincing, but I do agree that mid-high is the null hypothesis.
Allen thinks the fact that Latin ĭ was often transliterated as Greek ε indicates not that Greek ε sounded like an “i”, but that Latin ĭ sounded like an “e” (“peculiarly open”—which I’ll translate into IPA as [ɪ]—”and so as near to Greek ε as to ι”. Ditto Greek ο used for Latin ŭ (presumably closer to [ʊ]).
The fact that Greek o transliterated the ŭ of other languages, e.g. Persian and Sanskrit (Mardonius, Greek <Mardonios>, was Old Persian Marduniya), indicates to him that the alternative υ was not by then [u] at all but [y]. Greeks at this point are likely asking “then why didn’t they use ου?! <Mardounios>” Because ου was a long vowel (Mardūnios); and back in Classical times, the difference between short and long vowels was extremely important.