Look into hard fescue or sheep fescue. It's a dark-green, extremely-fine-bladed grass that under dry conditions will only get a few inches tall. I had it in my
yard in the desert, wonderful stuff that never needed mowing. Not sure how well it would hold up to high traffic, but if it gets even a little water it will fill in nicely and become plush rather than tall.
There are fine-bladed, low-growing varieties of Bermuda grass that are used on golf courses; might be another prospect. Bermuda does need a rotary mower if you ever plan to cut it.