Preflop
4X 3-bet is a bit large in position but certainly acceptable. You can mix it up with 88 and sometimes call, sometimes 3-bet.
Flop
C-bet for value is of course completely standard with bottom set, and I also like just calling his check-raise. You have position and two bets left, so you can just take over the betting, if he check the turn. I am very happy about getting raised here, since you only lose to TT. AA would almost always have put in a cold 4-bet.
Turn and river
You also lose to AT now, but if he can have AT, he can also have AJ, AQ and maybe AK as well, and those hands might still be betting for value. In top of that every single draw missed, so he can certainly have some bluffs. Folding a full house in this situation would be way to nitty.
Conclusion
Set over set is usually just a big cooler, and this hand was no exception to that. If he have the 88, and you have the TT, he also lose all his money to you, as the board ran out.