For me, it was the beginning of what would be more than half a decade of strategic avoidance and social discomfort.

what would be more than half a decade of strategic avoidance and social discomfort.为名词性关系分句作介词of的宾语，这个分句相当于一个名词词组加定语从句；

the thing which would be more than half a decade of strategic avoidance and social discomfort.

主句为一般过去时，所以从句用过去将来时。

如果要用what was, 则主句的谓语就要改为过去完成时：

For me, it had been the beginning of what was more than half a decade of strategic avoidance and social discomfort.

what would be/what was也可以省去不用：

For me, it was the beginning of more than half a decade of strategic avoidance and social discomfort.

此时就没有了过去将来时中would的预测意义，而表示这五年多的情况实实在在发生了。