Ireland went down 22-10 in the Six Nations clash, and Best believes that his side missed a great opportunity to reach the final four.
"It will take a while to get over this. There's no doubt that this was a fantastic chance, we all knew it," said Best. "We all knew that topping our group would put us in a north-south hemisphere split and turn it into knockout Six Nations rugby.
"We got to the quarter-final confident that we could win, but failed to achieve that. Any time you get to that stage of a competition it's always an opportunity missed, unless you get to the final and win it.
"We're bitterly disappointed. Credit to Wales, but for us the emotion is very raw. It's just one of those things unfortunately. There's been a great feeling throughout the squad while we've been here and we're all very disappointed to be going home."
Wales's victory has earned them a semi-final showdown against France on Saturday.