Answer: The Gemara (Sukka 47a) writes that there is a machlokes as to what one should do on Shemini Atzeres in chutz laaretz when there is a safek as to whether to treat it as the seventh or eighth day of Sukkos. The Gemara concludes that one should not shake the lulav and esrog though one should sit in the sukka without reciting the beracha, leishev basukka (See Rambam, Sukka 6:13; Rema OC 668:1; Yechave Daas 2:76). Tosafos explains that as people sometimes sit in a sukka during the year, it is less obvious as to why they are doing so than shaking one’s lulav and esrog.
R’ Tzadok Hakohen wrote a sefer (Meshiv Tzedek) defending the practice of not eating in one’s sukka on Shemini Atzeres. Likewise, the Aruch Hashulchan (OC 668:4) justifies this minhag explaining that Tosafos’ rationale wouldn’t apply in particularly cold climates when people would only sit in a sukka for the mitzva. R’ Moshe Sternbuch (Moadim Uzemanim 1:92) challenges this, however, as most of the rishonim lived in Europe and Russia and they never suggested that people should be exempt because of this.
There is a machlokes however, as to whether one should sleep in the sukka on Shemini Atzeres or not. While the Vilna Gaon (Maaseh Rav 222) writes that one should, the Mishna Berura (668:6) concludes that the minhag is not to.
In conclusion, one should eat in a sukka on Shemini Atzeres without reciting the beracha, leishev basukka, unless one has a specific minhag not to do so.