The shortest book of the Bible, it consists of one chapter with 21 verses.
There was a prophet Obadiah in the days of Elijah, during the reign of Jehoram of Judah (mid 9th century BC - 2 Ki 8:20-22; 2 Chron. 21:8-20). But this is not the same Obadiah, and the book doesn't make this claim. The internal evidence puts the book in the 6th century BC as it mainly condemns Edom for attacking Judah in the wake of the Babylonian incursion.
The reference to "house of Jacob" in this book specifically refers to Judah (1:17a, 1:19-21, especially v.21; cf. Jer. 5:20), so verse 10 doesn't condemn an Edomite attack on the northern kingdom. In verse 18, the dual reference to "house of Jacob" and "house of Joseph" denotes all Israel (cf. Zechariah 10:6). Seeing the similarity between vv.1-9 and Jer. 49:7-22, the idea that Obadiah wrote some time around Jeremiah's day is probably correct.
One can only imagine the rage of the author, in whose view kinsmen opportunists (Edomites) attacked his countrymen in their moment of weakness, the small kingdom already ravaged by the greed of the Babylonian giant. On the other hand, the Edomites had suffered vassalage under the Israelites for centuries, and although they weren't plundered, it certainly contributed to feelings of revanchism and irredentism.