jmc

Notice that once $n>8,$ the units digit of $n!!$ will be $0$ because there will be a factor of $10.$ Thus, we only need to calculate the units digit of $$2!!+4!!+6!!+8!!=2+8+48+48\cdot 8.$$ We only care about units digits, so we have $2+8+8+8\cdot 8,$ which has the same units digit as $2+8+8+4.$ The answer is $\boxed{2}.$