Take Borel measure on $[0,1]$ as an example. Cut off disjoint intervals $I_1,I_2,\dots$ where $I_i$ has length $2^{-i}\epsilon$.  That's length $\epsilon$ altogether. In the remaining $1-\epsilon$, take independent events $B_1,B_2,\ldots$ with $B_i$ of measure $(1-2^{-i})\epsilon$.  Define $A_i=I_i\cup B_i$.  Then $A_i$ has measure $\epsilon$ for all $i$, and $A_i\cap A_j$ has measure $(1-2^{-i})(1-2^{-j})\epsilon^2\lt \epsilon^2$ for $i\ne j$.

Do I need to cite a theorem that $B_1,B_2,\ldots$ exist? I think it is standard elementary probability, and anyway it is easy to prove using finite unions of intervals.